Itohan Osayi - Assigment #1

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Itohan Osayi MA-311-OL8 Mathematical Modeling For Business Applications 03/01/2024 Exercises 1.3 Determine if the equations below are linear. If an equation is linear, rearrange it into the form y = mx+b and give the values of m and b. 1. 2x + 5x – y = 9 Combining like terms: 7x − y = 9 Rearrange: −y = −7x + 9 ( linear ) Multiply by -1 to isolate y: y = 7x − 9 So, it is linear with m = 7 and b = −9 2. 3xy – 2x = y x ( 3y-2) = y ( not linear ) This equation is not linear because it contains a product of x and y. 3. x = -3y – 8y + 3x – 9 3y - 8y = 3x - x - 9 11y/11 = 2x /11 - 9/11 Y = 2/11x - 9/11 ( linear ) m = 2/11 b = -9/11
Combining like terms: −2x = −11y −9 Divide by -2 to isolate y: y = 11/2x + 9/2 So, it is linear with m = 11/2 and b = 9/2 4. –3/x = 5/y Cross-multiply to get rid of fractions: −3y = 5x Rearrange: y = −5/3x ( linear ) So, it is linear with m = −5/3 and b = 0 5. 4x^2 + 2y – 3x = 4x^2 2y = 4x^2 - 4x^2 + 3x 2y = 3x Y = 3/2x ( linear ) m = 3/2 , b = 0 This equation is linear, but when simplified, 2y=3x, so it is not uniquely defined in the form y=mx+b. 6. -2y = -8 – 3y -2y + 3y = -8 y = -8 ( linear ) m = 0 b = -8 Combine like terms: −2y + 3y = −8 Simplify: y = −8 ( linear ) So, it is linear with m = 0 and b = −8 7. x^3/x^2 = 5 - y Simplify: x = 5 − y Rearrange: y = −x + 5 ( linear ) So, it is linear with
m = −1 and b = 5 In summary: ● Equations 1, 3, 4, 5, 6, and 7 are linear. ● Equation 2 is not linear. Exercises 1.4 For each of the equations below create a table of 5 ordered pairs and, using any two of these pairs, calculate the ratio of the change in y to the change in x. 1. y = 7x – 5 x y 0 -5 1 2 2 9 3 16 4 23 5 30 Ratio: 16 − 9 7 —— = —
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3 - 2 1 2. y = 4x +7 x y 0 7 1 11 2 15 3 19 4 23 5 27 Ratio: 19 − 15 4 —— = — 3 - 2 1 3. y = -4x + 5 x y 0 5 1 1 2 -3
3 -7 4 -11 5 -15 Ratio: -7 − (-3) -4 —— = — 3 - 2 1 4. y = 1.5x + 3 x y 0 3 1 4.5 2 6 3 7.5 4 9 5 10.5 Ratio: 7.5 - 6 1.5 —— = — 3 - 2 1
5. y = 7x/3 + 1/2 x y 0 1 1 8 2 15 3 22 4 29 5 36 Ratio: 22 - 15 7 —— = — 3 - 2 1 6. y = -1.87x + 3.91 x y 0 3.91 1 2.04 2 0.17
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3 -1.7 4 -3.57 5 -5.44 Ratio: -1.7 - 0.17 1.87 ———— = –— 3 - 2 1 7. 4y + 5x = 9 Equation: 4y+5x=9 Rearranged to 9 - 5x y = ——— 4 x y 0 2.25 1 1 2 -0.25 3 -1.5 4 -2.75
5 -4 Ratio: -1.5 - (-0.25) -1.25 ———— = –— 3 - 2 1 8. 3(2x –4) + 3y = 3x Equation: 3(2x − 4) + 3y = 3x (rearranged to y = −2/3x + 4 ) x y 0 4 1 3.33 2 2.67 3 2 4 1.33 5 0.66 Ratio: 2- 2.67 -0.67 ———— = –— 3 - 2 1 9. 2x + 3y = 2y – 7x Equation: 2x+3y=2y−7x Rearranged to
x y 0 0 1 -9 2 -18 3 -27 4 -36 5 -45 Ratio: -27 - (-18) -9 ————— = –— 3 - 2 1 10. y = x x y 0 0 1 1 2 2 3 3 4 4
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Ratio: 3 - 2 1 —— = –— 2 - 1 1 Exercise 1.5 For each of the situations given in problems 1 through 8 provide the following: Please provide the following: a. Definition or meaning of x, including the units of x. b. Definition or meaning of y, including the units of y c. The equation in the form of y = mx +b. d. Two ordered pairs that satisfy the equation. 1. Long distance phone service costs $5.00 per month with a usage fee of 8 cents per minute. The equation you create will allow you to calculate the total monthly phone cost from the number of minutes the phone is used. a. Definition of x: The number of minutes the phone is used. ● Units of x: Minutes b. Definition of y: The total monthly phone cost. ● Units of y: Dollars c. Equation in the form of y = mx + b: ● The fixed monthly cost ($5.00) represents the y-intercept (b), and the usage fee per minute (8 cents) represents the slope (m). So, the equation becomes: ● y = 0.08x+5.00
d. Two ordered pairs that satisfy the equation: For x = 50 minutes: y = 0.08(50) + 5.00 y = 4.00 + 5.00 y = 9.00 So, when the phone is used for 50 minutes, the total monthly phone cost is $9.00. For x = 100 minutes: y = 0.08(100) + 5.00 y =8.00 + 5.00 y = 13.00 So, when the phone is used for 100 minutes, the total monthly phone cost is $13.00. Therefore, two ordered pairs that satisfy the equation are (50,9.00) and (100,13.00) 2. The cost to rent a car is $185.00 per week and $.40 per mile. The equation you create will allow you to use the mileage driven in a week to calculate the total weekly rental cost. a. Definition of x: The mileage driven in a week. ● Units of x: Miles b. Definition of y: The total weekly rental cost.
● Units of y: Dollars c. Equation in the form of y = mx + b: ● The fixed weekly cost ($185.00) represents the y-intercept (b), and the cost per mile ($0.40) represents the slope (m). So, the equation becomes: y=0.40x+185.00 d. Two ordered pairs that satisfy the equation: ● x (mileage driven in a week) into the equation and solve for y (total weekly rental cost). ● For example: For x=200 miles: y=0.40(200)+185.00 y=80.00+185.00 y=265.00 So, when driving 200 miles in a week, the total weekly rental cost is $265.00. For x=300 miles: y=0.40(300)+185.00 y=120.00+185.00 y=305.00 So, when driving 300 miles in a week, the total weekly rental cost is $305.00. Therefore, two ordered pairs that satisfy the equation are (200,265.00) and (300,305.00)
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3. The value of a piece of equipment depreciates linearly. It cost $135,000 new. Each year it depreciates at a rate of 10% of its original purchase price. The equation you create will relate the value of the equipment to its age. (Hint: the slope in negative ) a. Definition of x: The age of the piece of equipment in years. ● Units of x: Years b. Definition of y: The current value of the equipment. ● Units of y: Dollars c. Equation in the form of y = mx + b: ● The initial cost of the equipment ($135,000.00) represents the y-intercept (b), and the depreciation rate (10% per year) represents the slope (m). Since depreciation is a decrease in value, the slope will be negative. ● The depreciation rate of 10% can be represented as 0.10 in decimal form. So, the equation becomes: y=−0.10x+135,000.00 d. Two ordered pairs that satisfy the equation: ● For example: For x=1 year (1 year of depreciation): y=−0.10(1)+135,000.00 y=−13,500+135,000.00 y=121,500.00
So, when the equipment is 1 year old, its value is $121,500.00. For x=5 years (5 years of depreciation): y=−0.10(5)+135,000.00 y=−67,500+135,000.00 y=67,500.00 So, when the equipment is 5 years old, its value is $67,500.00. Therefore, two ordered pairs that satisfy the equation are (1,121,500.00) and (5,67,500.00) 4. The amount of water used by a hotel when there are no guests is 2000 gallons per day. Each guest uses, on average, 150 gallons per day. The equation you create will relate total water use per day to the number of guests in the hotel for that day. a. Definition of x: The number of guests staying at the hotel for a given day. ● Units of x: Number of guests b. Definition of y: Total water use per day by the hotel. ● Units of y: Gallons c. Equation in the form of y = mx + b:
● The fixed amount of water used by the hotel when there are no guests (2000 gallons per day) represents the y-intercept (b), and the water use per guest (150 gallons per day) represents the slope (m). So, the equation becomes: y=150x+2000 d. Two ordered pairs that satisfy the equation: ● To find ordered pairs, we'll plug in different values for x (number of guests) into the equation and calculate the corresponding y (total water use per day). ● For example: For x=0 guests (when there are no guests): y=150(0)+2000 y=0+2000 y=2000 So, when there are no guests at the hotel, the total water use per day is 2000 gallons. For x=10 guests: y=150(10)+2000 y=1500+2000 y=3500 So, when there are 10 guests at the hotel, the total water use per day is 3500 gallons. Therefore, two ordered pairs that satisfy the equation are
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(0,2000) and (10,3500) 5. The cost for electric service is a $6.50 per month connection charge plus $.15 per kilowatt-hour. The equation you create will allow you to predict the total monthly charge from the amount of electricity used in that month a. Definition of x: The amount of electricity used in a month. ● Units of x: Kilowatt-hours (kWh) b. Definition of y: The total monthly charge for electric service. ● Units of y: Dollars c. Equation in the form of y = mx + b: ● The fixed monthly connection charge ($6.50) represents the y-intercept (b), and the cost per kilowatt-hour ($0.15) represents the slope (m). So, the equation becomes: y=0.15x+6.50 d. Two ordered pairs that satisfy the equation: We'll plug in different values for x (amount of electricity used in a month) into the equation and calculate the corresponding y (total monthly charge). For example: For x=50 kWh: y=0.15(50)+6.50 y=7.50+6.50 y=14.00
So, when 50 kWh of electricity is used in a month, the total monthly charge is $14.00. For x=100 kWh: y=0.15(100)+6.50 y=15.00+6.50 y=21.50 So, when 100 kWh of electricity is used in a month, the total monthly charge is $21.50. Therefore, two ordered pairs that satisfy the equation are (50,14.00) and (100,21.50) 6. Archaeologists working at a dig site estimate that every inch of depth represents 500 years of age. The equation you create will relate the age of a layer to its depth. a. Definition of x: The depth of the layer at the archaeological dig site. ● Units of x: Inches b. Definition of y: The age of the layer at the archaeological dig site. ● Units of y: Years c. Equation in the form of y = mx + b: ● The relationship between depth and age can be expressed as a linear equation, where the slope represents the rate of change in age with respect to depth. ● Since every inch of depth represents 500 years of age, the slope m is 500. ● However, we need to determine the y-intercept (b) from the information given. Without specific information, we can't determine the exact value of the y-intercept. It depends on
where the depth is measured from (e.g., surface level, sea level, etc.) or if there are any specific initial conditions provided. So, the equation will be: y=500x+b d. Two ordered pairs that satisfy the equation: A specific ordered pair can’t be provided without knowing the value of the y-intercept (b). The y-intercept represents the age of the layer at a particular depth (e.g., the age of the layer at the surface). However, if the y-intercept can be 0 (meaning the age at the surface is 0), then we can find ordered pairs based on that assumption. For example: ● At x=0 (depth at the surface), if we assume y=0 (age at the surface), then the ordered pair is (0,0) ● At x=1 inch (depth of 1 inch), the age would be y=500(1)+0=500 years. So, the ordered pair is (1,500) ● At x=2 inches (depth of 2 inches), the age would be y=500(2)+0=1000 years. So, the ordered pair is (2,1000). These ordered pairs represent hypothetical scenarios based on the assumption of a y-intercept of 0. Actual ordered pairs would depend on the specific context and information provided about the dig site. 7. The slope of a roof is expressed as its pitch. A 4 pitch roof increases vertically 4 inches for every foot the roof extends horizontally. If the lowest part of a 4 pitch roof is 14 feet above the ground, create an equation that relates the height of the roof to horizontal distance from the lowest point. (Hint: be careful of the units)
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a. x: Horizontal distance from the lowest point of the roof, measured in feet. b. y: Height of the roof above the ground, measured in feet. c. Equation in the form of y = mx + b: To express the relationship between the height of the roof and the horizontal distance from the lowest point, we'll use the equation of a line, where: ● m: Slope of the roof, which is the pitch. Given that the pitch is 4, the slope (m) is 4. ● b: The y-intercept, which is the height of the lowest point of the roof. Given that the lowest part of the roof is 14 feet above the ground, the y-intercept (b) is 14. Therefore, the equation becomes: y=4x+14 d. Two ordered pairs that satisfy the equation: ● When x=0 (meaning right at the lowest point of the roof), y=4(0)+14=14. So, the ordered pair is (0, 14). ● When x=3 (three feet away horizontally), y=4(3)+14=12+14=26. So, the ordered pair is (3, 26). So, two ordered pairs that satisfy the equation are (0, 14) and (3, 26). 8. It is estimated that 40,000 acres of farmland are lost to development in New Jersey each year. If there were approximately 800,000 acres of active farmland in the state in 2006, create an equation that relates the amount of farmland left to the number of years that have elapsed since 2006. a. x: Number of years that have elapsed since 2006.
b. y: Amount of farmland left in acres. c. Equation in the form of y = mx + b: Since 40,000 acres are lost each year, the slope (m) of the equation would be -40,000 (negative because it represents a decrease). The y-intercept (b) would be the initial amount of farmland, which is 800,000 acres in 2006. Therefore, the equation becomes: y=−40,000x+800,000 d. Two ordered pairs that satisfy the equation: ● When x=0 (year 2006), y=−40,000(0)+800,000=800,000. So, the ordered pair is (0, 800,000). ● When x=1 (1 year after 2006), y=−40,000(1)+800,000=760,000. So, the ordered pair is (1, 760,000). So, two ordered pairs that satisfy the equation are (0, 800,000) and (1, 760,000). For problems 9 through 15 calculate the slope using two points and the equation: m = (y2 – y1 ) / (x2 – x1 ) 9. y = 3x – 4 This equation is already in slope-intercept form (y=mx+b). The slope (m) is 3. 10. y = 5x/2 + 2 This equation is in slope-intercept form. The slope (m) is 5/2 11. y = -2x – 3 The slope (m) is -2. 12. y = -3x/4 - ½
The slope (m) is −3/4 13. 3x – 2y = y + 7 To calculate the slope, let's rearrange the equation into slope-intercept form y=mx+b: 3x−2y=y+7 3x=3y+7 3y=3x−7 y= 3/3x - 7/3 The slope (m) is 1. 14. 4 = 8x-6/y 4y=8x−6 y= 8/4x - 6/4 y=2x-3/2 The slope (m) is 2. 15. y = 3.2x – 4.1 The slope (m) is 3.2.
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Exercises 1.6 Graph the Following Equations 1. y = 2x +4 2. y = -3x – 3 3. y = 4x/3 + 7
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4. y = 5x/6 – ¾ 5. y = 1.3x + 3.4
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6. 2y + 6 = -6x + 4 2y = −6x − 2 2y = −6x − 2 y = −3x − 1 y = −3x −1 7. –2x – y = x – 6 y = −2x −x + 6 y =−2x −x + 6 y = −3x + 6
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y = −3x + 6 8. 3y -4x = -2x + 5 3y = 2x −4x +5 3y = −2x + 5 y = -2/3x + 5/3 9. y-2(3x -3) = -6x + 4.5 y − 6x + 6 = −6x + 4.5 y = −6x + 4.5 − 6 y = −6x − 1.5
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10. 3y = -2x + 4 y = −2/3x + 4/3 Exercises 1.7 Find the Equation of the Straight Line that goes through both given points or that has the given point and slope. 1. (1, 3) (0, 1)
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● Slope m=3−1/1−0=2/1=2 ● Using point-slope form: y−3=2(x−1) ● After simplification: y = 2x + 1 2. (2, 10) (1, 6) ● Slope m=10−6/2−1=4/1=4 ● Using point-slope form: y−10=4(x−2) ● After simplification: y = 4x + 2 3. (-2, 12) (4, 9) ● Slope m= 9-12/4-(-2) = -3/6 = -½ ● Using point-slope form: y−12= -½ (x + 2) ● ( Distribute -½ ) y−12= -½x + 1 ● Move −12: y = -1/2x - 1 + 12 ● After simplification: y = -1/2x + 11 4. (-2, 12) (2, -8) ● Slope m = -8-12/2-(02) = -20/4 = -5 ● Using point-slope form: y -12 = -5 (x-(-2)) ● y-12=-5(x+2) ● y-12=-5x-10 ● y=-5x+2 ● After simplification: y = -5x+2 5. (-2, 1) (0, -5) ● Slope m = -5 - 1 / 0 - (-2) = -6/2 = -3 ● Using point-slope form: y -1 = -3 (x-(-2)) ● y-1=-3(x+2) ● y-1=-3x-6
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● y=-3x-5 ● After simplification: y = -3x-5 6. (4, –2) (0, 4) ● Slope m = 4-(-2)/0-4=6/-4=-3/2 ● Using point-slope form: y - (-2)=-3/2(x-4) ● y+2=-3/2(x-4) ● y+2=-3/2x+6 ● y=-3/2x+4 ● After simplification: y = -3/2x+4 7. (5, -2) (-1, -2) ● Slope m = -2-(-2)/-1-5=0 ● Using point-slope form: y - (-2) = 0(x-5) ● y+2=0 ● y = -2 ● After simplification: y = -2 8. (-1, -6) (2, 12) ● Slope m = 12-(-6)/2-(-1)=18/3=6 ● Using point-slope form: y - (-6) = 6(x-(-1) ● y+6=6(x+1) ● y+6=6x+6 ● y=6x ● After simplification: y = 6x 9. (6, 5) (2, 2) ● Slope m = 2-5/2-6 = -3/-4 = ¾ ● Using point-slope form: y-5=¾(x-6) ● y-5=3/4x-9/2
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● y=3/4x-9/2+5 ● y=3/4x+1/2 ● After simplification: y = 3/4x+1/2 10. (6, 3 ¾) (-1, -11/12) ● Slope m = -11/12 - ¾ / -1 - 6 ● ¾ and -11/12 common denominator is 12 ● m = -11/12 - 9/12 / -7 ● m = -20/12 / -7 ● Simplify: ● m = -5/3/-7 = 5/21 ● Using point-slope form: y - ¾ = 5/21(x-6) ● y - ¾ = 5/21x - 30/21 ● y = 5/21x - 30/21 + ¾ ● y = 5/21x - 30/21 + 63/84 ● y = 5/21x - 30/21 + 63/84 ● y = 5/21x - 10/21 ● After simplification: y = 5/21x - 10/21 11. Slope = 0 (1, -10) If the slope (m) is given as 0, it means the line is horizontal, and the equation of the line is in the form y=b, where b is the y-intercept. Given the point (1, -10) on the line, the equation would be y = −10. So, the equation of the straight line is y = −10 12. Slope = 2/3 (0, 8) If the slope (m) is given as 2/3 and a point on the line is (0, 8), the point-slope form of the equation would be:
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y - 8 = ⅔ (x - 0 ) Simplify: y - 8 = 2/3x y = 2/3x + 8 The equation of the straight line is y = 2/3x + 8 Exercise 1.8 1. Your long distance phone service has a base monthly charge and a per- minute charge. When you used 350 minutes in a month the total cost was $32.50. When you used 400 minutes in a month the total cost was $36.50. You want an equation that will allow you to calculate your phone bill. Please provide: a. the definition of x, including the units b. the definition of y, including the units
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c. the equation in the form y = mx + b d. the units of the slope e. the units of b and an interpretation of b. f. How much will your phone bill be if you talked for 711 minutes? a . Definition of x: ● x represents the number of minutes used in a month. ● Units: minutes b. Definition of y: ● y represents the total cost of the phone bill. ● Units: dollars c. Equation in the form y = mx + b: ● m = per-minute charge and b = base monthly charge. ● The equation is: x = 350, y = $32.50 and when x = 400, y = $36.50, ● Equations: ● m = 36.50-32.50/400-350 = 4/50 = 0.08 dollars/minute ● 32.50=0.08×350+b ● b=32.50−0.08×350=32.50−28=4.50 dollars ● So, the equation becomes: y=0.08x+4.50 d. Units of the slope (m): The units of the slope are dollars per minute ($/minute).
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e. Units of b and Interpretation of b: The units of b are dollars ($). b represents the base monthly charge, which is the fixed cost of the service irrespective of the number of minutes used. f . To find the phone bill if you talked for 711 minutes: y=0.08×711+4.50 y=56.88+4.50=61.38 dollars So, the phone bill would be $61.38 if you talked for 711 minutes. 2. The dosage for a medicine is linear with the weight of the patient. There is a minimum dosage onto which is added a per pound dosage. You find that your dosage, at 110 pounds is 48 milliliters (ml). Your brother’s dosage, at 170 pounds, is 66 ml. You would like an equation that will relate the dosage to the weight of the patient. Please provide: a. the definition of x, including the units b. the definition of y, including the units c. the equation in the form y = mx + b
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d. the units of the slope e. the units of b and an interpretation of b. f. What is the dosage for a 165 lb patient? a. Definition of x: ● x: Weight of the patient ● Units: Pounds (lbs) b. Definition of y: ● y: Dosage of the medicine ● Units: Milliliters (ml) c. Equation in the form y = mx + b: ● Given two points (110, 48) and (170, 66), we can find the slope m and the y-intercept b using the point-slope form of the linear equation: ● m=y2−y1/x2−x1 ● b=y−mx Substituting the given points: m=66−48/170−110=18/60=0.3 Using one of the points to find 48=0.3×110+b
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b=48−33=15 Thus, the equation becomes: y=0.3x+15 d. Units of the slope ( m ): Since dosage is given in ml and weight in pounds, the units of the slope (m) will be in ml per pound (ml/lb). e. Units of b and an interpretation of b represents the y-intercept, the dosage when the weight is 0 pounds. It could indicate a minimum dosage or a base dosage that every patient receives regardless of their weight. f. Dosage for a 165 lb patient: ● Using the equation y=0.3x+15 plug in x=165: ● y=0.3×165+15=49.5+15=64.5 ml ● So, the dosage for a 165 lb patient would be 64.5 ml. 3. You work in a manufacturing plant on a machine that requires time for set-up and a time to make each part. The machine can make 100 parts in 38 minutes. The machine can make 175 parts in one hour, four minutes and fifteen seconds. You would like an equation that relates the number of parts made to the time it takes to make the parts.
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Please provide: ( Hint: Be careful of the units.) a. the definition of x, including the units b. the definition of y, including the units c. the equation in the form y = mx + b d. the units of the slope e. the units of b and an interpretation of b. f. how long would it take to make 500 parts? a. Definition of x: ● x: Time to make the parts ● Units: Minutes b. Definition of y: ● y: Number of parts made ● Units: Parts c. Equation in the form y = mx + b: ● Given that the machine can make 100 parts in 38 minutes and 175 parts in one hour, four minutes, and fifteen seconds, we can use these points to find the specific equation. Convert the second time point to minutes:
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● 1 hour = 60 minutes ● 4 minutes + 15 seconds = 4 + (15/60) = 4.25 minutes So, 1 hour, 4 minutes, and 15 seconds is 60 + 4 + 0.25 = 64.25 minutes. We have two points: (38, 100) and (64.25, 175). To find the equation, we need to find the slope (m) and y-intercept (b) using these points. m=y2−y1/x2−x1 = 175−100/64.25−38 m = 75/26.25 ≈2.857 100=2.857×38+b b=100−(2.857×38) b≈100−108.446=−8.446 Thus, the equation becomes: y=2.857x−8.446 d. Units of the slope (m): ● Since y is the number of parts and x is the time in minutes, the units of the slope (m) will be parts per minute (parts/min). e. Units of b and an interpretation of b: ● b represents the y-intercept, which in this context would be the initial number of parts made when the time is 0. It might not have a straightforward physical interpretation in
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this context since making negative parts doesn't make sense. It's more about the model's initial adjustment. f. How long would it take to make 500 parts? ● To find the time (x) it takes to make 500 parts, we plug y=500 into the equation y=2.857x−8.446 and solve for x. 500=2.857x−8.446 2.857x=500+8.446 2.857x=508.446 x=508.446/2.857≈178.009 minutes So, it would take approximately 178.009 minutes to make 500 parts. 4. A piece of machinery depreciates linearly. It’s worth $120,000 when it’s 2 years old. It’s worth $90,000 when it’s 4 years old. You want to create an equation that relates the age of the machine to its value. Please provide: a. the definition of x, including the units b. the definition of y, including the units
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c. the equation in the form y = mx + b d. the units of the slope e. the units of b and an interpretation of b. f. how much would the machine be worth when its 8 years old? a. Definition of x: ● x: Age of the machine ● Units: Years b. Definition of y: ● y: Value of the machine ● Units: Dollars c. Equation in the form y = mx + b: ● Given that the machine's worth is $120,000 when it's 2 years old and $90,000 when it's 4 years old, we can use these points to find the specific equation. Using the point-slope form of the linear equation: m=y2−y1/x2−x1 b=y−mx Substitute the values: m=90,000−120,000/4−2=−30,000/2=−15,000
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120,000=−15,000×2+b b=120,000+30,000=150,000 Thus, the equation becomes: y=−15,000x+150,000 d. Units of the slope (m): Since y is the value of the machine and x is the age in years, the units of the slope (m) will be dollars per year ($/year). e. Units of b and an interpretation of b: ● b represents the y-intercept, which in this context would be the value of the machine when it is 0 years old. In this case, it doesn't have a practical interpretation since the machine can't be aged 0 years. The value of b is $150,000. f. How much would the machine be worth when it's 8 years old? ● To find the value (y) when the machine is 8 years old, will need to plug x=8 into the equation y=−15,000x+150,000: ● y=−15,000×8+150,000=120,000 So, the machine would be worth $120,000 when it's 8 years old. 5. Assume that the amount of tread remaining on a tire decreases linearly with the tire’s mileage. After 30,000 miles there is .61 inches of tread left.
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After 50,000 miles there is a quarter of an inch of tread left. We want an equation that relates the mileage on a tire to the number of inches of tread left. Please provide: a. the definition of x, including the units b. the definition of y, including the units c. the equation in the form y = mx + b d. the units of the slope e. the units of b and an interpretation of b. f. how much tread would be left on a tire that has traveled 60,000 miles? a. Definition of x: ● x: Mileage on the tire ● Units: Miles b. Definition of y: ● y: Amount of tread remaining on the tire ● Units: Inches c. Equation in the form y = mx + b:
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● Given that there is 0.61 inches of tread left after 30,000 miles and a quarter of an inch of tread left after 50,000 miles, we can use these points to find the specific equation. Using the point-slope form of the linear equation: m=y2−y1/x2−x1 b=y−mx Substitute the values: m=0.25−0.61/50,000−30,000 m=−0.36/20,000=−0.000018 0.61=−0.000018×30,000+b b=0.61+0.54=1.15 Thus, the equation becomes: y=−0.000018x+1.15 d. Units of the slope (m): ● Since y is the amount of tread in inches and x is the mileage in miles, the units of the slope (m) will be inches per mile (in/mile). e. Units of b and an interpretation of b: b represents the y-intercept, which in this context would be the amount of tread left when the mileage is 0 miles. It might not have a practical interpretation since a tire can't have negative mileage. The value of b is 1.15 inches.
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f. How much tread would be left on a tire that has traveled 60,000 miles? ● To find the amount of tread (y) when the tire has traveled 60,000 miles, plug x=60,000 into the equation y=−0.000018x+1.15 ● y=−0.000018×60,000+1.15=0.14 So, there would be 0.14 inches of tread left on a tire that has traveled 60,000 miles. 6 –10. For each of the problems, 6 – 10, cost information at a certain level of production for a manufacturing process is given. The revenue per unit is also given. Assume a linear relationship between the number of units produced and cost. For each problem please find: a. The cost, revenue and profit functions and the units of the slope and of b. b. The variable cost per unit, also known as the marginal cost, and the fixed cost. c. The cost, revenue and profit when z units are produced. (z will be specified in each problem) d. The break-even point. e. The average cost per unit of producing w units and the equation of the
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average cost per unit function. (w will be specified in each problem) 6. The cost of producing 1,250 units is $45,000. The cost of producing 1,500 units is $50,000. Each unit produced can be sold for $30. z = 3,000; w = 2500. Part a: Cost Function (C): C(x)=mx+b Given that the cost of producing 1,250 units is $45,000 and the cost of producing 1,500 units is $50,000: C(1250)=45000 C(1500)=50000 Slope (m) and y-intercept (b): m=50000−45000/1500−1250 = 5000/250=20 Substitute one of the points to find b:
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45000=20×1250+b b=45000−25000=20000 So, the cost function is: C(x)=20x+20000 Revenue Function (R): R(x)=px Given that each unit can be sold for $30: R(x)=30x Profit Function (P): P(x)=R(x)−C(x) P(x)=(30x)−(20x+20000) P(x)=10x−20000 Part b: Variable Cost per Unit (Marginal Cost) (m):
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● The slope (m) of the cost function is the marginal cost. ● m=20 Fixed Cost (b): ● The y-intercept (b) of the cost function is the fixed cost. ● b=20000 Part c: Cost when z units are produced (C(z)): C(3000)=20×3000+20000=80000 Revenue when z units are produced (R(z)): R(3000)=30×3000=90000 Profit when z units are produced (P(z)): P(3000)=10×3000−20000=10000 Part d: Break-even Point: ● Break-even occurs when P(x)=0. ● 10x−20000=0
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● 10x=20000 ● x=2000 So, the break-even point is 2000 units. Part e: Average Cost per Unit of Producing w units (AC(w)): AC(x)=C(x)/x AC(2500)=20×2500+20000/2500=70000/2500=28 So, the average cost per unit when producing 2500 units is $28. The equation of the average cost per unit function is: AC(x)=28 7. The cost of producing 23,000 units is $225,000. The cost of producing 30,000 units is $260,000. Each unit produced can be sold for $12. z = 45,000; w = 68,000. Part a: Cost Function ( C ): C(x)=mx+b
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Given that the cost of producing 23,000 units is $225,000 and the cost of producing 30,000 units is $260,000: C(23000)=225000 C(30000)=260000 The slope (m) and y-intercept (b): m=260000−225000/30000−23000=35000/7000=5 Now, substitute one of the points to find b: 225000=5×23000+b b=225000−115000=110000 So, the cost function is: C(x)=5x+110000 Revenue Function ( R ): R(x)=px Given that each unit can be sold for $12: R(x)=12x Profit Function ( P ): P(x)=R(x)−C(x)
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P(x)=(12x)−(5x+110000) P(x)=7x−110000 Part b: Variable Cost per Unit (Marginal Cost) ( m) : The slope (m) of the cost function is the marginal cost. ● m=5 Fixed Cost ( b ): ● The y-intercept (b) of the cost function is the fixed cost. ● b=110000 Part c: Cost when z units are produced ( C(z) ): C(45000)=5×45000+110000=335000 Revenue when z units are produced ( R(z) ): R(45000)=12×45000=540000 Profit when z units are produced ( P(z) ):
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P(45000)=7×45000−110000=215000 Part d: Break-even Point: ● Break-even occurs when P(x)=0 ● 7x−110000=0 ● 7x=110000 ● x=15714.29 So, the break-even point is approximately 15,714 units. Part e: Average Cost per Unit of Producing w units ( AC(w) ): AC(x)=C(x)/x AC(68000)=5×68000+110000/68000 AC(68000)=450000/68000 AC(68000)≈6.618 So, the average cost per unit when producing 68,000 units is approximately $6.62. The equation of the average cost per unit function is: AC(x)≈6.618
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8. The cost of producing 11,000 units is $89,000. The cost of producing 15,000 units is $95,000. Each unit produced can be sold for $6. z = 23,000; w = 35,000. a. Cost, Revenue, and Profit Functions: ● C(x) = Cost function ● R(x) = Revenue function ● P(x) = Profit function ● x = number of units produced and sold Two points: (11,000, $89,000) and (15,000, $95,000). Slope, m=95000−89000/15000−11000=6000/4000=1.5 Intercept, b=89000−(1.5×11000)=89000−16500=72500 So, the cost function is: C(x)=1.5x+72500 The revenue function is straightforward, as each unit sold brings in $6: R(x)=6x Profit function:
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P(x)=R(x)−C(x)=6x−(1.5x+72500)=4.5x−72500 b. Variable Cost per Unit (Marginal Cost) and Fixed Cost: Variable Cost (Marginal Cost): This is the coefficient of x in the cost function, which is $1.5 per unit. Fixed Cost: This is the y-intercept of the cost function, which is $72,500. c. Cost, Revenue, and Profit when z=23,000 units are produced: ● Cost: C(23000)=1.5(23000)+72500=34500+72500=107000 ● Revenue: R(23000)=6(23000)=138000 ● Profit: P(23000)=R(23000)−C(23000)=138000−107000=31000 d. Break-even Point: The break-even point is where the profit function equals zero: P(x)=0 4.5x−72500=0 4.5x=72500 x=72500/4.5 x≈16111.11
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So, the break-even point is approximately 16,111 units. e. Average Cost per Unit of Producing w=35,000 units: Average Cost per Unit is the total cost divided by the number of units produced. Average Cost = Total Cost / Number of Units Average Cost = C(w)/w Average Cost = 1.5×35000+72500/35000 Average Cost = 125000/35000 Average Cost ≈ $3.57 per unit. So, the equation of the average cost per unit function is simply $3.57 per unit. 9. The cost of producing 200,000 units is $4,445,181. The cost of producing 300,000 units is $5,164,181. Each unit produced can be sold for $11.29. z = 250,000; w = 318,000.
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a. Cost, Revenue, and Profit Functions: Let's denote: ● C(x) = Cost function ● R(x) = Revenue function ● P(x) = Profit function ● x = number of units produced and sold We are given two points: (200,000, $4,445,181) and (300,000, $5,164,181). Using the two points, we can find the slope and intercept of the cost function to get the cost equation. Slope, m=5164181−4445181/300000−200000=719000/100000=7.19 Intercept, b=4445181−(7.19×200000)=4445181−1438000=3007181 So, the cost function is: C(x)=7.19x+3007181 The revenue function is straightforward, as each unit sold brings in $11.29: R(x)=11.29x Profit function: P(x)=R(x)−C(x)=11.29x−(7.19x+3007181)=4.1x−3007181 b. Variable Cost per Unit (Marginal Cost) and Fixed Cost:
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Variable Cost (Marginal Cost): This is the coefficient of x in the cost function, which is 7.19 dollars per unit. Fixed Cost: This is the y-intercept of the cost function, which is $3,007,181. c. Cost, Revenue, and Profit when z=250,000 units are produced: ● Cost: C(250000)=7.19(250000)+3007181=1797500+3007181=4804681 ● Revenue: R(250000)=11.29(250000)=2822500 ● Profit: P(250000)=R(250000)−C(250000)=2822500−4804681=−1982181 d. Break-even Point: The break-even point is where the profit function equals zero: P(x)=0 4.1x−3007181=0 4.1x=3007181 x=3007181/4.1 x≈733676.34 So, the break-even point is approximately 733,676 units. e. Average Cost per Unit of Producing w=318,000 units: Average Cost per Unit is the total cost divided by the number of units produced. Average Cost = Total Cost / Number of Units Average Cost = C(w)/w
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Average Cost = 7.19×318000+3007181/318000 Average Cost ≈ 2291242+3007181/318000 Average Cost ≈ 5298423/318000 Average Cost ≈ $16.67 per unit. So, the equation of the average cost per unit function is approximately $16.67 per unit. 10. The cost of producing 22,575,300 units is $4,235,502.20. The cost of producing 31,897,650 units is $5,857,591.10. Each unit produced can be sold for $.23. z = 23,455,000; w = 36,573,500. a. Cost, Revenue, and Profit Functions ● C(x) as the total cost function ● R(x) as the total revenue function ● P(x) as the total profit function For linear cost, revenue, and profit functions, we can use the slope-intercept form: ● Cost function: C(x)=mx+b ● Revenue function: R(x)=px ● Profit function: P(x)=R(x)−C(x)
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We need to find the slope (m) and the y-intercept (b). Given two points: (22,575,300,$4,235,502.20) and (31,897,650,$5,857,591.10) We can find the slope using the formula: m=change in cost/change in units = $5,857,591.10−$4,235,502.20/31,897,650−22,575,300 m=$1,622,088.90/9,322,350≈0.1739 Points to find b (22,575,300,$4,235,502.20) $4,235,502.20=(0.1739)(22,575,300)+b b=$4,235,502.20−(0.1739)(22,575,300) b≈$790,627.77 So, the cost function is: C(x)=0.1739x+790,627.77 The revenue function is straightforward: R(x)=0.23x
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The profit function is: P(x)=R(x)−C(x)=(0.23x)−(0.1739x+790,627.77) P(x)=0.0561x−790,627.77 b. Variable Cost per Unit and Fixed Cost ● Variable cost per unit (marginal cost): It's the coefficient of x in the cost function. In this case, it's the slope, which is $0.1739. ● Fixed cost: It's the y-intercept of the cost function, which is $790,627.77. c. Cost, Revenue, and Profit when z units are produced Given z=23,455,000: ● Cost: C(z)=0.1739(23,455,000)+790,627.77 ● Revenue: R(z)=0.23(23,455,000) ● Profit: P(z)=R(z)−C(z) You can calculate these values using the formulas above. d. Break-even Point The break-even point is where the total revenue equals the total cost, or where P(x)=0. 0=0.0561x−790,627.77 0.0561x=790,627.77
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x=790,627.77/0.0561≈14,083,150 So, the break-even point is approximately 14,083,150 units. e. Average Cost per Unit of Producing w Units Given w=36,573,500: Average cost per unit is the total cost divided by the number of units produced. Average cost per unit=C(w)/w Substitute x=36,573,500 into the cost function: C(36,573,500)=0.1739(36,573,500)+790,627.77 Now, calculate the average cost per unit: Average cost per unit=0.1739(36,573,500)+790,627.77/36,573,500 Average cost per unit=0.1739×36,573,500+790,627.77/36,573,500 Average cost per unit≈6,354,332.65+790,627.77/36,573,500 Average cost per unit≈7,144,960.42/36,573,500 Average cost per unit≈0.1956 So, the average cost per unit of producing w=36,573,500 units is approximately $0.1956. Use the following information for problems 11 and 12. There are two main temperature scales in common use. In the United State the Fahrenheit Scale is often used. In this scale water freezes at approximately 32 degrees and boils at approximately 212 degrees. In the Celsius Scale, which is used in countries using the metric system and is also used for many
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applications in the US, water freezes at approximately 0 degrees and boils at approximately 100 degrees. The two scales are related linearly. 11. Determine the equation that allows you to calculate the temperature in Fahrenheit degrees from the temperature in Celsius degrees. What are the units of the slope? Given: ● Fahrenheit freezing point: 32 ● Fahrenheit boiling point: 212 ● Celsius freezing point: 0 ● Celsius boiling point: 100 Equation for converting Celsius to Fahrenheit: The relationship between Fahrenheit (F) and Celsius (C) is linear. The general form of a linear equation is y=mx+b, where m is the slope and b is the y-intercept. We can use the two points (0,32) and (100,212) to find the slope (m): m=change in F/ change in C=212−32/100−0 m=180/100=1.8 So, the equation for converting Celsius to Fahrenheit is: F=1.8C+32
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Units of the Slope: The slope m in this case is 1.8. The units of the slope are degrees Fahrenheit per degree Celsius. 12. Determine the equation that allows you to calculate the temperature in Celsius degrees from the temperature in Fahrenheit degrees. What are the units of the slope? Given: ● Fahrenheit freezing point: 32 ● Fahrenheit boiling point: 212 ● Celsius freezing point: 0 ● Celsius boiling point: 100 Equation for converting Fahrenheit to Celsius: Similarly, we can use the two points (32,0) and (212,100) to find the slope (m): m = change in C/change in F=100−0/212−32 m=100/180=5/9 So, the equation for converting Fahrenheit to Celsius is:
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C=5/9(F−32) Units of the Slope: The slope m in this case is 5/9. The units of the slope are degrees Celsius per degree Fahrenheit. EXERCISES 2.2 Create a table of ordered pairs and graph the following exponential functions. 1. f(x) = 3^x For x=−2:f(−2)=3^-2= 1/9 For x=−1:f(−1)=3^-1=1/3 For x=0:f(0)=3^0=1 For x=1:f(1)=3^1 For x=2:f(2)=3^2=9 x f(x)
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-2 1/9 -1 1/3 0 1 1 3 2 9
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2. f(x) = 3(2.3)^2x For x=−2:f(−2)=3(2.3)^−4 ≈6.3 For x=−1:f(−1)=3(2.3)^−2 ≈13.8 For x=0:f(0)=3(2.3)^0=30
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For x=1:f(1)=3(2.3)^2 ≈65.4 For x=2:f(2)=3(2.3)^4 ≈142.2 x f(x) -2 6.3 -1 13.8 0 30 1 65.4 2 142.2
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3. g(x) = 1.7(1.04)^2x-1 For x=−2:g(−2)=1.7(1.04)^−5 ≈5.6356 For x=−1:g(−1)=1.7(1.04)^−3 ≈3.38 For x=0:g(0)=1.7(1.04)^−1≈1.7
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For x=1:g(1)=1.7(1.04)^1 ≈1.856 For x=2:g(2)=1.7(1.04)^3 ≈2.03824 x g(x) -2 5.6356 -1 3.38 0 1.7 1 1.856 2 2.03824
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4. c(x) = (1/2)^x For x=−2:c(−2)=(½)^-2=4 For x=−1:c(−1)=(½)^-1=2 For x=0:c(0)=(½)^0=1 For x=1:c(1)=(½)^1=½
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For x=2:c(2)=(½)^2=¼ x c(x) -2 4 -1 2 0 1 1 0.5 2 0.25
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5. t(x) = 3.7(.93)^2x For x=−2:t(−2)=3.7(0.93)^−4 ≈15.9778 For x=−1:t(−1)=3.7(0.93)^−2 ≈29.7
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For x=0:t(0)=3.7(0.93)^0 =3.7 For x=1:t(1)=3.7(0.93)^2 ≈6.8964 For x=2:t(2)=3.7(0.93)^4 ≈12.8604 x t(x) -2 15.9778 -1 29.7 0 3.7 1 6.8964 2 12.8604
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6. r(x) = 3 + .87^-2x For x=−2:r(−2)=3+0.87^−2(−2)≈3.1 For x=−1:r(−1)=3+0.87^−2(−1≈3.87
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For x=0:r(0)=3+0.87^0=4 For x=1:r(1)=3+0.87^−2(1)≈3.8715 For x=2:r(2)=3+0.87^−2(2)≈3.1 x r(x) -2 3.1 -1 3.87 0 4 1 3.8715 2 3.1
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7. r(x) = 2^x-2 For x=−2:r(−2)=2^−2-2=−3/4 For x=−1:r(−1)=2^−1−2=−1.5 For x=0:r(0)=2^0−2=0
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For x=1:r(1)=2^1−2=0 For x=2:r(2)=2^2−2=2 x r(x) -2 -2.25 -1 -1 0 0 1 1 2 2
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8. f(x) = e^x For x=−2:f(−2)=e^−2≈0.1353 For x=−1:f(−1)=e^−1≈0.3679 For x=0:f(0)=e^0=1
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For x=1:f(1)=e^1≈2.7183 For x=2:f(2)=e^2 ≈7.3891 x f(x) -2 0.1353 -1 0.3679 0 1 1 2.7183 2 7.3891
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9. f(x) = e^-.67x For x=−2:f(−2)=e^−(−0.67×−2)≈2.6745 For x=−1:f(−1)=e^−(−0.67×−1)≈1.7956
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For x=0:f(0)=e^−(−0.67×0)=1 For x=1:f(1)=e^−(−0.67×1)≈0.6738 For x=2:f(2)=e^−(−0.67×2)≈0.4515 x f(x) -2 2.6745 -1 1.7956 0 1 1 0.6738 2 0.4515
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10. f(x) = 2e^2-x For x=−2:f(−2)=2e^2−(−2) =14.7781 For x=−1:f(−1)=2e^2−(−1) ≈7.3891 For x=0:f(0)=2e^2−0 =2 For x=1:f(1)=2e^2−1 ≈0.8109
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For x=2:f(2)=2e^2−2 =0.1353 x f(x) -2 14.7781 -1 7.3891 0 2 1 0.8109 2 0.1353
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For problems 11 to 15 round the answer to the hundredths place. 11. Use your calculator to evaluate log 109.3 log(109.3)≈2.04 12. Use your calculator to evaluate log 79.3
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log(79.3)≈1.90 13. Use your calculator to evaluate log 4.3 log(4.3)≈0.63 14. Use your calculator to evaluate ln 21.1 ln(21.1)≈3.05 15. Use your calculator to evaluate ln 2.718281828 ln(2.718281828)≈1.00 16. Evaluate without a calculator ln e^2.1 In(e^2.1)=2.1 (Because ln (e^x)=x) For problems 17 to 20 indicate whether the table of ordered pairs could be generated from an equation that is linear, exponential or neither. 17. x y -5 13 -3 7 -2 4
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0 -2 1 -5 6 -20 Looking at the table, we see that the differences between consecutive y values do not remain constant as x increases or decreases. Therefore, it's neither linear nor exponential. 18. x y -4 0.4 -1 1.3 0 2.0 2 4.5 5 15.2 6 22.8 The differences between consecutive x values are not constant, so it's neither linear nor exponential. 19.
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x y -3 5.1 -1 1.8 0 1.1 1 0.7 2 0.4 3 0.2 The differences between consecutive y values decrease as x increases. This suggests that it could be an exponential function. 20. x y -5 32.0 -3 14.0 0 2.0 2 4.0 4 14.0 5 22.0 The pattern of y values doesn't seem to follow a consistent exponential growth or decay. It's neither linear nor exponential. So, to summarize: Neither linear nor exponential. Neither linear nor exponential. Possibly exponential. Neither linear nor exponential.
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Exercise 2.3 1. The population of Zimbabwe was 12.7 million in 2004. The growth rate is 1.2% per year. What would you expect the population to be in the year 2100? The population of Zimbabwe in 2004 was 12.7 million. To find the population in 2100, we need to calculate the growth over 96 years. Using the formula for exponential growth: P(t)=P0×(1+r)^t where: ● P(t) is the population after time t, ● P0 is the initial population (12.7 million), ● r is the growth rate (1.2% per year), ● t is the time in years. P(96)=12.7×(1+0.012)^96 P(96)≈12.7×(1.012)^96 P(96)≈12.7×3.979 P(96)≈50.553 million So, the population of Zimbabwe in the year 2100 is expected to be approximately 50.553 million. 2. Panama had a population of 3.2 million in 2004. How long would it take for the population to grow to 5 million? The growth rate is 1.8% per year. P(t)=P0×(1+r)t where:
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● P(t) is the population after time t, ● P0 is the initial population (3.2 million), ● r is the growth rate (1.8% per year), ● t is the time in years. 5=3.2×(1+0.018)t 5=3.2×(1+0.018) Solving for t: (1.018)t=53.2 t≈ln (5/3.2)/ln (1.018) t≈ln (1.5625)/ln (1.018) t≈0.4447/0.0173 t≈25.67 So, it would take approximately 25.67 years for Panama's population to grow to 5 million. 3. Haiti had a population of 8.1 million in 2004. If the population is expected to grow to 20.0 million by 2070, what is the growth rate? For Haiti, to find the growth rate, we can use the formula for exponential growth and solve for P(t)=P0×(1+r)t where: ● P(t) is the population after time t, ● is the initial population (8.1 million), ● r is the growth rate, ● t is the time in years. Given
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P(t)=20.0 million and t= years, we need to solve for r: 20.0=8.1×(1+r)^t Since we don't have the value of t, we cannot directly solve for r. 4. Venezuela’s population is expected to be 35.3 million in 2027. The growth rate is 1.9% per year. What was Venezuela’s population in 2006? To find Venezuela's population in 2006, we need to backtrack from the expected population in 2027. We can use the formula for exponential growth: P(t)=P0×(1+r)^t Rearranging for P0 P0=P(t)/(1+r)^t Given: ● P(t)=35.3 million (2027) ● r=1.9% per year ● t=2027−2006=21 years P0=35.3/(1+0.019)^21 P0≈35.3/(1.019)^21 P0≈35.3/2.034 P0≈17.35 million So, Venezuela's population in 2006 was approximately 17.35 million. 5. The population of the world is estimated to be 6.396 billion. The growth rate is 1.3% per year. How long before the population of the world doubles?
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To find out how long it will take for the world population to double, we use the formula for exponential growth and solve for t: P(t)=P0×(1+r)^t Given: ● P0=6.396 billion ● r=1.3% per year ● P(t)=2×6.396=12.792 billion ● 12.792=6.396×(1+0.013)^t Solving for t: (1.013)t=12.792/6.396 t≈ln (2)/ln (1.013) t≈0.693/0.0128 t≈54.14 years So, it would take approximately 54.14 years for the world population to double. 6. The population of the world is estimated to be 6.396 billion and grows at a rate of 1.3% per year. How long before the population grows by 75%? To find out how long it will take for the world population to grow by 75%, we use the formula for exponential growth and solve for P(t)=P0×(1+r)^t Given: ● P0==6.396 billion ● r=1.3% per year ● P(t)=6.396×(1+0.013×0.75)=6.396×1.00975=6.44 billion
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6.44=6.396×(1+0.013)^t Solving for t (1.013)t=6.446.396 t≈ln (1.00975)/ln (1.013) t≈0.00972/0.0128 t≈0.758 years So, it would take approximately 0.758 years for the world population to grow by 75%. 7. Canada’s population growth rate is .3% per year. If the population of Canada was estimated to be 31.9 million in 2004, what will it be in 2054? To find Canada's population in 2054, we use the formula for exponential growth: P(t)=P0×(1+r)t Given: ● P0=31.9 million (2004) ● r=0.3% per year ● t=2054−2004=50 years P(50)=31.9×(1+0.003)^50 P(50)≈31.9×(1.003)^50 P(50)≈31.9×1.165 P(50)≈37.17 million So, Canada's population in 2054 is expected to be approximately 37.17 million. 8. What is the growth rate of a population that doubles in 35 years?
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The growth rate of a population that doubles in t years can be found using the formula for exponential growth: P(t)=P0×(1+r)t Given: ● P(t)=2×P0 ● P(t)=P0×(1+r)^t Substituting 2×P0 for P(t) in the second equation: 2×P0=P0×(1+r)^t Dividing both sides by P0 2=(1+r)^t Taking the logarithm of both sides: log (2)=log ((1+r)^t) log (2)=t×log (1+r) Solving for r r=10 log (2)/^t−1 Given: t=t=35 years r=10log (2)/35−1 r≈0.0198 So, the growth rate is approximately 1.98% per year. 9. China has a population of 1.3 billion today. With a growth rate of .6% per year how long will it be before the population grows by 700,000,000? To find out how long it will take for China's population to grow by 700,000,000, we can use the formula for exponential growth: P(t)=P0×(1+r)^t
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Given: ● P0=1.3 billion ● r=0.6% per year ● ΔP=700,000,000 We want to find t such that P(t)=P0+ΔP P(t)=1.3+0.7=2 2=1.3×(1+0.006)^t Solving for t (1.006)^t=21.3 t≈ln (2/1.3)/ln (1.006) t≈ln (1.53846)/ln (1.006) t≈0.4297/0.00599 t≈71.67 years So, it would take approximately 71.67 years for China's population to grow by 700,000,000 10. With a growth rate of .6% per year how long before the population of the US doubles? To find out how long it will take for the population of the US to double with a growth rate of 0.6% per year, we use the formula for exponential growth: P(t)=P0 x (1+r)^t Given: ● P0=1 (initial population) ● r=0.6% per year ● P(t)=2 (double the initial population)
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2=1×(1+0.006)t Solving for t (1.006)t=2 t≈ln (2)/ln (1.006) t≈0.693/0.00599 t≈115.77 years So, it would take approximately 115.77 years for the population of the US to double 11. Bulgaria, like a number of Eastern European countries, has a negative population growth rate. If Bulgaria’s growth rate is -.6% per year and its population is 7.8 million in 2006 what will the population be in 2050? To find Bulgaria's population in 2050, we use the formula for exponential growth: P(t)=P0 ×(1+r)^t Given: ● P0=7.8 million (2006) ● r=−0.6% per year ● t=2050−2006=44 years P(44)=7.8×(1−0.006)^44
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P(44)≈7.8×(0.994)^44 P(44)≈7.8×0.616 P(44)≈4.81 million So, Bulgaria's population in 2050 is expected to be approximately 4.81 million. 12. What is the doubling time of a population that grows by 80% in 45 years? To find the doubling time of a population, we can use the concept of exponential growth and the formula for doubling time: Doubling Time=ln (2)/ln (1+r) Where r is the growth rate expressed as a decimal. For a population that grows by 80% in 45 years, we first need to find the growth rate r. The growth rate can be calculated using the formula: r=P(t)−P0/P0 Where: ● P(t) is the final population (doubled population in this case), ● P0 is the initial population. Given:
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● Initial population P0=1 (assuming 100% as base), ● Final population P(t)=1+0.8=1.8 (80% increase). ● r = 1.8-1/1 = 0.8 Doubling Time=ln (2)/ln (1+0.8) Doubling Time≈0.693/ln (1.8) Doubling Time≈0.693/0.5878 Doubling Time≈1.18 years So, the doubling time for a population that grows by 80% in 45 years is approximately 1.18 years. 13. What is the doubling time of a population that grows from 34.5 million to 54.5 million in 23 years? For a population that grows from 34.5 million to 54.5 million in 23 years, we first need to find the growth rate r. The growth rate can be calculated using the formula: r = P(t)−P0/P0 Given: Initial population P0=34.5 million, Final population P(t)=54.5 million, Time t=23 years.
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r=54.5−34.5/34.5 r≈20/34.5 r≈0.5797 Doubling Time=ln (2)/ln (1+0.5797) Doubling Time≈0.693/ln (1.5797) Doubling Time≈0.693/0.4544 Doubling Time≈1.526 years So, the doubling time for a population that grows from 34.5 million to 54.5 million in 23 years is approximately 1.526 years.
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