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Average Rate of Change of a Function In Exercises 61-64, find the average rate of change of the function from
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- Disappearing Mothball The rate at which the volume of a mothball evaporates from solid to gas is proportional to the surface area of the ball. Suppose a mothball has been observed at one time to have a radius of 0.5 in. and, six months later, a radius of 0.25 in. (a) Express the radius of the mothball as a function of time. (b) When will the mothball disappear completely?arrow_forwardExercise refer to the functionarrow_forwardGasoline mileage Suppose the cost of driving an automo- bile is a linear function of the number x of miles driven and that gasoline costs $3 per gallon. A certain automobile presently gets 20 mi/gal, and a tune-up that will improve gasoline mileage by 10% costs $120. (a) Express the cost C, of driving without a tune-up in terms of x. (b) Express the cost C, of driving with a tune-up in terms of x. (c) How many miles must the automobile be driven after a tune-up to make the cost of the tune-up worthwhile?arrow_forward
- 34) From a full 50-liter container of a 40% concentration of acid, x liters is removed and replaced with 100% acid. (a) Write the amount of acid in the final mixture as a function of x.arrow_forwardArea Painted (ft) The graph shows the linear relationship between the maximum area in square feet that can be painted and the number of gallons of paint used. 9 Paint Coverage 1,800 1,600 1,400 8 1,200 E 1,000 800 600 400 200 1 3 Number of Gallons of Paint Which of these best represents the rate of change of the maximum area painted with respect to the number of gallons of paint used? 200 ft²/gal 1 ft2/gal 200 B 400 ft?/gal 1 ft?/gal 400 O A O Carrow_forwardDistance between cars At noon, car A is 10 feet to the right and 20 feet ahead of car B, as shown in the figure. If car A continues at 88 ft/sec (or 60 mi/hr) while car B continues at 66 f/sec (or 45 mi/hr), express the distance d between the cars as a function of t, where t denotes the number of sec- onds after noon. Exercise 78arrow_forward
- Pipe is to be laid from point A in a rectangular plot of size 10 x 20 feet to a point P on side BC and from there on to C as shown in red in the figure below. The cost of laying pipe within the plot is $c per foot (it must be underground), while the cost of laying pipe along the side of the plot is $d per foot. 10 A 20 (a) Determine the total cost, f(x), as a function of c, d and x, where x is the distance from P to B. (Note that f(x) is discontinuous at x = 0.) f(x) = dollars (b) What is the cost of laying the pipe on the side of the plot (x = 0) when c = 30 and d = 25? (Round your answer to the nearest cent.) (c) What is the least cost of laying the pipe when c = 30 and d = 25? (Round your answer to the nearest cent.)arrow_forwardof a product under relatively stable market conditions but in the absence of promotional activities such as advertisingtend to decline at a constant yearly rateThis rate of saarrow_forwardWave height as a function of wind speed and duration Duration (hours) Wind speed (knots) V 10 15 20 30 40 50 60 5 2 4 5 10 14 2 4 7 9 13 15 2 5 8 16 21 25 19 29 36 24 37 47 20 30 40 50 2 5 5 8 17 28 40 54 (a) What is the value of f(30, 20)? What is its meaning? According to the table, f(30, 20) = feet. 2 I 9° 18 31 45 62 2 LO 5 9 33 48 2 19 19 67 5 9 33 50 69 which means that if a 30 knot wind has been blowing in the open sea for 20 hours, it will create waves with estimated heights of (b) What is the meaning of the function of h = f(50, t)? Describe the behavior of this function. We fix v and t, resulting in a constant value. We fix v = 50 and allow t to vary, resulting in an equation of one variable. We fix t = 50 and allow v to vary, resulting in an equation of one variable. We allow v and t to vary, resulting in a function of two variables. (c) What is the meaning of the function h = f(v, 50)? Describe the behavior of this function. We fix v and t, resulting in a constant value. We…arrow_forward
- World grain production was 1,241 million tons in 1975 and 2,048 million tons in 2005, and has been increasing at an approximately constant rate. (a) Find a linear function for world grain production, P, in million tons, as a function of t, the number of years since 1975.arrow_forwardFill in the blanks: The derivative of a product of two functions is the first function times the of the second plus the second function times the derivative of the first. ( Previous Continue >arrow_forwardBoyle's Law 2 states that for a certain gas in a container we have P. V = 350 where P represents the pressure of the gas (in mmHG) and V represents the volume of the gas (in liters). a. If the pressure of the gas is 240 mmHG, what is the volume of the gas? liters Preview b. Write a function f that determines the volume of the gas (in liters) in terms of the pressure of the gas in mmHG, P. f(P) Preview c. Complete the following statement. (Hint: if f(P) increases without bound, enter "oo". If f(P) decreases without bound, enter "- o0".) As P → 0*, f(P) → Previewarrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
- College AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage