Written Project #1 Version B - Itohan Osayi
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School
Fashion Institute Of Technology *
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Course
322
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
20
Uploaded by GrandSalmonPerson1036
Written Project #1 Version B
Name: Itohan Osayi
1. Determine if the equations below are linear: (5 points)
a)
???
= −
𝟗?
−
??
+
𝟕?
−
?
This equation is linear because each term is raised to the power of 1 and there are no terms where
variables are multiplied together.
b)
???
+
??
−
??
=
???
This equation is not linear because it contains a term with
x
raised to the power of 4, violating the
criteria for linearity. Therefore, the equation is not linear.
2. The dosage for a medicine is linear with the weight of patient. There is a minimum
dosage onto which is added a per pound dosage. You find that your dosage, at 125 pounds
is 46 milliliters (ml). Your brother’s dosage, at 150 pounds, is 66 ml. You would like an
equation that will relate the dosage to the weight of the patient. (10 points)
a. Find the equation in the form of y=m*x + b.
b. Find the units of b and an interpretation of b.
c. What is the dosage for a 129 lb patient?
a. To find the equation relating dosage to the weight of the patient in the form of
y
=
mx
+
b
, where
y
is the dosage (in ml),
x
is the weight of the patient (in pounds),
m
is the per
pound dosage, and
b
is the minimum dosage, we can use the given data points (125 pounds, 46
ml) and (150 pounds, 66 ml).
Using the point-slope formula
m=y2−y1/x2−x1, where (x1,y1) , (x2,y2) are the given points:
m=66−46/150−125=20/25=0.8
Now, we can use one of the points and
m
to find
b
. We can use the point (125 pounds, 46 ml):
46=0.8×125+b
46=100+b
b
=46−100=−54
So, the equation relating dosage to the weight of the patient is
y
=0.8
x
−54.
b. The unit of
b
is milliliters (ml).
b
represents the minimum dosage that is added regardless of
the patient's weight. In this case,
b
=−54 ml, meaning that there is a minimum dosage of 54 ml
that is added.
c. To find the dosage for a 129 lb patient, we can plug
x
=129 into the equation:
y=0.8×129−54
y
=103.2−54
y
=49.2
So, the dosage for a 129 lb patient would be 49.2 ml.
3. Find the equation of the Straight Line that goes through both given points: (5 points)
(-36,-2) & (0, 7)
To find the equation of the straight line that goes through the given points
(−36,−2) and (0,7), we can use the point-slope form of the equation of a line:
y−y1=m(x−x1)
Where (x1,y1) is one of the given points, and
m
is the slope of the line.
The slope (
m
) using the given points:
m=y2−y1/x2−x1
Substitute the coordinates of the points
(−36,−2) and (0,7):
m=7−(−2)/0−(−36) m=7+2/0+36
m=9/36
m=1/4
Now that we have the slope m=1/4 , we can use one of the given points, for example,
(−36,−2) and substitute into the point-slope form:
y−(−2)=1/4(x−(−36)) y+2=1/4(x+36)
Now, let's simplify:
y+2=1/4x+1/4×36
y+2=1/4x+9
Subtract 2 from both sides:
y=1/4x+9−2 y=1/4x+7
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So, the equation of the straight line that goes through the points
(−36,−2) and (0,7) is y=1/4x+7
4. Consider a manufacturing plant which produces 280,000 units for $350,000 and 560,000
units for $420,000. Each unit can be sold for $3.6 (15 points)
a. Find the cost, revenue, and profit functions.
b. Find the cost of producing 365,000 units.
c. What is the break-even point?
To find the cost, revenue, and profit functions, we need to understand the cost, revenue, and
profit for each unit produced.
●
C
(
x
) as the cost function, where
x
is the number of units produced.
●
R
(
x
) as the revenue function, where
x
is the number of units sold.
●
P
(
x
) as the profit function, where
x
is the number of units sold.
We're given two data points:
Producing 280,000 units costs $350,000.
Producing 560,000 units costs $420,000.
From these, we can find the cost per unit:
Cost per unit = Total cost / Number of units produced
Cost per unit = $350,000 / 280,000 = $1.25 per unit
Cost per unit = $420,000 / 560,000 = $0.75 per unit
a. Cost Function:
●
Since the cost is linearly related to the number of units produced, we can use the equation
of a line to find the cost function.
●
Given the points (280,000, $350,000) and (560,000, $420,000), we can find the slope
(cost per unit) and the y-intercept (fixed cost).
●
Slope m=420,000−350,000/560,000−280,000=70,000/280,000=0.25
●
Using point-slope form:
●
C(x)=0.25x+b
●
Substituting one of the points, say (280,000, $350,000), we get:
●
350,000=0.25×280,000+
b
●
b
=350,000−0.25×280,000=350,000−70,000=280,000
●
So, the cost function is
C
(
x
)=0.25
x
+280,000.
Revenue Function:
●
Revenue = Price per unit × Number of units sold
●
Revenue = $3.6 × Number of units sold
●
R(x)=3.6x
Profit Function:
●
Profit = Revenue - Cost
●
P(x)=R(x)−C(x)=3.6x−(0.25x+280,000)=3.35x−280,000
b. Cost of producing 365,000 units:
●
C(365,000)=0.25(365,000)+280,000
●
C(365,000) = $91,250 + $280,000
●
C(365,000) = $371,250
c. Break-even point:
●
At the break-even point, profit is zero.
●
So,
P
(
x
)=0
●
3.35
x
−280,000=0
●
3.35
x
=280,000
●
x=280,0003.35
●
x
≈83,582.09
So, the break-even point is approximately 83,582 units.
5. Your long distance phone service has a base monthly charge and a per-minute charge.
When
you used 375 minutes in a month the total cost was $48.50. When you used 450 minutes in a
month the total cost was $56.00. You want an equation that will allow you to calculate your
phone
bill. Please provide: (10 points)
a. The equation in the form y = mx + b
b. The units of the slope
c. The units of b and an interpretation of b.
d. How much will your phone bill be if you talked for 988 minutes?
a. To find the equation that allows you to calculate your phone bill, we'll use the given
information to form a linear equation in the form
y
=
mx
+
b
, where:
●
y
represents the total cost of the phone bill.
●
x
represents the number of minutes used.
●
m
represents the per-minute charge.
●
b
represents the base monthly charge.
Given data points:
When 375 minutes were used, the total cost was $48.50.
When 450 minutes were used, the total cost was $56.00.
First, let's find the slope (
m
) using the two data points:
m=56.00−48.50/450−375
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m=7.50/75
m
=0.10
Now, let's use one of the data points to find the value of
b
. We'll use the point (375 minutes,
$48.50).
48.50=0.10×375+
b
48.50=37.50+
b
b
=48.50−37.50
b
=11.00
So, the equation that allows you to calculate your phone bill is:
y
=0.10
x
+11.00
b. The units of the slope (
m
) are dollars per minute ($/min). In this case, ( m = 0.10 $/min.
c. The units of
b
are dollars ($).
b
represents the base monthly charge, which is the cost incurred
regardless of the number of minutes used. In this case,
b
=$11.00. This means that even if you
didn't use any minutes, you would still have to pay $11.00.
d. To find out how much your phone bill will be if you talked for 988 minutes, you can use the
equation y=0.10x+11.00
y=0.10×988+11.00
y=98.80+11.00
y=109.80
So, your phone bill will be $109.80 if you talked for 988 minutes.
6. Create a table of ordered pairs and graph the following exponential functions. (10 points)
1.
𝒇
(
?
) =
?
.
??
2.
𝒇
(
?
) =
?
∗
??
3
For the first function
f(x)=1.5x
Let's choose
x
=−2,−1,0,1,2 and calculate
f
(
x
) for each:
f(−2)=1.5^−2=1/1.5^2=1/2.25=0.444
f(−1)=1.5^−1=1/1.5=0.667
f(0)=1.5^0=1
f(1)=1.5^1=1.5
f(2)=1.5^2=2.25
So, the ordered pairs for the first function are:
(−2,0.444), (−1,0.667), (0,1), (1,1.5), (2,2.25)
x
f(x)
-2
0.444
-1
0.667
0
1
1
1.5
2
2.25
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For the second function
f(x)=3×2^x
We can choose the same
x
values as before and calculate
f
(
x
) for each:
f(−2)=3×2^−2=3×1/2^2=3×1/4=0.75
f(−1)=3×2^−1=3×1/2=1.5
f(0)=3×2^0=3×1=3
f(1)=3×2^1=3×2=6
f(2)=3×2^2=3×4=12
So, the ordered pairs for the second function are:
(−2,0.75), (−1,1.5), (0,3), (1,6), (2,12)
x
f(x)
-2
0.75
-1
1.5
0
3
1
6
2
12
7. The population of Almenia was 14.6 million in 1994. The growth rate is 1.6% per year.
What would you expect the population to be in the year 2060? (10 points)
To find the expected population of Almenia in the year 2060, we can use the formula for
exponential growth:
P(t)=P0×(1+r)t
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P
(
t
)=
P
0
×(1+
r
)
t
Where:
●
P
(
t
) is the population at time
t
.
●
P0 is the initial population (in 1994).
●
r
is the growth rate per year (in decimal form).
●
t
is the time in years since the initial population.
Given:
●
P0=14.6 million
●
r
=1.6%=0.016 (as a decimal)
●
t
=2060−1994=66 years (from 1994 to 2060)
Now, let's plug in these values into the formula:
P
(66)=14.6×(1+0.016)^66
P(66)=14.6×(1.016)^66
Using a calculator, we get:
P(66)≈14.6×(1.016)^66≈14.6×2.667≈39.02
So, we would expect the population of Almenia to be approximately 39.02 million in the year
2060.
8. Benin had a population of 7.2 million in 2012. If the population is expected to grow to
12.4 million by 2060, what is the growth rate? (10 points)
To find the growth rate of Benin's population, we can use the formula for exponential growth:
P(t)=P0×(1+r)^t
Where:
●
P
(
t
) is the population at time t
●
P0 is the initial population (in 2012).
●
r
is the growth rate per year (in decimal form).
●
t
is the time in years since the initial population.
Given:
●
P0=7.2 million
●
P
(2060)=12.4 million
●
t
=2060−2012=48 years (from 2012 to 2060)
We need to solve for
r
using the given values. Rearranging the formula, we have:
(1+r)^48=P(2060)/P0=12.4/7.2
Now, let's solve for
r
:
(1+r)^48=12.4/7.2
Taking the 48th root of both sides:
1+r=(12.4/7.2)^1/48
1+r=(12.4/7.2)^1/48≈1.00872
r≈1.00872−1
r
≈0.00872
So, the growth rate of Benin's population is approximately 0.872% per year.
9. What is the tripling time of a population that grows from 24.9 million to 36.8 million in
27 years? (10 points)
The tripling time of a population refers to the time it takes for the population to triple in size. We
can find this time using the formula for exponential growth:
P(t)=P0×(1+r)^t
Where:
●
P
(
t
) is the population at time
t
.
●
P0 is the initial population.
●
r
is the growth rate per year (in decimal form).
●
t
is the time in years.
Given:
●
P0=24.9 million
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●
P
(27)=36.8 million
●
We want to find the time it takes for the population to triple (P(t)=3P0
We can set up the equation:
3P0=P0×(1+r)^t
Since we're given the population after 27 years, we can use these values to solve for the growth
rate (
r
):
36.8=24.9×(1+r)^27
First, isolate (1+r)^27
36.8/24.9=(1+r)^27
Now, can find the 27th root of both sides:
(36.8/24.9)^1/27=1+r
(36.8/24.9)^1/27≈1.0143
1+
r
≈1.0143
r
≈1.0143−1
r
≈0.0143
Now, we know the growth rate (
r
) is approximately 0.0143 per year.
To find the tripling time, we need to find the time it takes for the population to triple from
24.9 million to 36.8 million. Since the population triples, we need to solve for
t
in the equation:
3×24.9=24.9×(1+0.0143)^t
Divide both sides by24.9:
74.7/24.9×(1.0143)^t
3=(1.0143)t
Now, let's take the natural logarithm (ln) of both sides to solve for
t
:
ln
(3)=ln
((1.0143)^t)
ln(3)=
t
×ln(1.0143)
Now, divide both sides by ln
(1.0143)
t
= ln(3)/ln(1.0143)
Now, let's calculate this:
t≈1.0986/0.0142 t
≈77.38
So, the tripling time of the population is approximately 77.38 years.
10. What is the doubling time of a population that grows by 60% in 15 years? (15 points)
The doubling time of a population refers to the time it takes for the population to double in size.
We can find this time using the formula for exponential growth:
P(t)=P0×(1+r)^t
Where:
●
P
(
t
) is the population at time
t
.
●
P0 is the initial population.
●
r
is the growth rate per year (in decimal form).
●
t
is the time in years.
Given:
●
The population grows by 60% in 15 years.
To find the growth rate (
r
), we can use the percentage growth formula:
Percentage growth=P(t)−P0/P0×100
Given that the population grows by 60% over 15 years, we have:
60%=P(t)−P0/P0×100
0.60=P(t)−P0/P0
Now, we can use the given values to find
r
:
0.60=P(t)−P0/P0
0.60=P(t)/P0−1
1.60=P(t)/P0
P(t)/P0=1.60
Given that the population doubles, we have:
P(t)/P0=2
Now, let's equate these two expressions:
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1.60=2
This implies that:
(1+r)^15=2
Now, we can solve for
r
:
(1+r)^15=2
Taking the 15th root of both sides:
1+r=2/15
r=2/15−1
Calculate
r
:
r≈0.03872
So, the growth rate (
r
) is approximately 0.03872.
To find the doubling time, we need to find the time it takes for the population to double. We can
use the formula:
2=(1+r)^t
Now, let's solve for
t
:
(1+0.03872)^t=2
1.03872^t=2
t=ln
(2)/ln
(1.03872)
Calculate:
t≈0.6931/0.0384
t
≈17.99
So, the doubling time of the population is approximately 17.99 years.