Reference: Calculus and Its Applications, 12th, by Marvin L. Bittinger and David Ellenbogen Book

Elementary Applied Calculus

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These are practice questions

Links to Book Below:

1. Calculus And Its Applications, Brief Version 12th edition | Print ISBN - 9780135164884, eText ISBN - 9780135225103 | VitalSource

2. Bittinger, Ellenbogen, Surgent & Kramer, Calculus and Its Applications: Brief Version, 12th Edition | Pearson

These are are practice questions

6. Maximize Q = xy, where x and y are positive numbers such that x + 3y* = 16. Write the objective function in terms of y- Q = "Type an expression using y as the variable.) The interval of interest of the objective function is (Simplify your answer. Type your answer in interval notation.) The maximum value of Q is (Simplify your answer.) 7. A lifeguard needs to rope off a rectangular swimming area in front of Long Lake Beach, using 700 yd of rope and floats. What dimensions of the rectangle will maximize the area? What is the maximum area? (Note that the shoreline is one side of the rectangle.) Let x be the length of a side of the rectangle perpendicular to the shoreline. Write the objective function for the area in terms of x. A(x) = (Type an expression using x as the variable.) The length of the shorter side of the rectangular region is (1) The length of the longer side of the rectangular region is (2) The maximum area of the rectangular region is (3) (1) O yd. (2) O yd. (3) O yd. O yd'. O yds. O yds. O yaz. O yd. 8. Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 43 in. by 23 in. by cutting congruent squares from the corners and folding up the sides. Then find the volume. The dimensions of box of maximum volume are (1). Round to the nearest hundredth as needed. Use a comma to separate answers as needed.) The maximum volume is (2) (Round to the nearest hundredth as needed.) (1) O in.. (2) O in." O in. in. O in. O in. . 9. A waste management company is designing a rectangular construction dumpster that will be twice as long as it is wide and must hold 26 yd" of debris. Find the dimensions of the dumpster that will minimize its surface area. Write the surface area formula in terms of the width, x. Assume the dumpster has an open top. SA = The width of the dumpster is (1) The length of the dumpster is (2) The height of the dumpster is (3) [Round to the nearest hundredth as needed.) (1) O yd. (2) O yd. (3) O yd. O yd3 O yd O yd? O yd3.10. Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, R(x), and cost, C(x), of producing x units are in dollars. R(x) = 40x - 0.1x", C(x) = 6x + 10 In order to yield the maximum profit of $_ units must be produced and sold. (Simplify your answers. Round to the nearest cent as needed.) 11. Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, R(x), and cost, C(x), are in thousands of dollars, and x is in thousands of units. R(x) = 7x - 3x". C(x) = x - 4x + 4x+1 The production level for the maximum profit is about units. (Do not round until the final answer. Then round to the whole number as needed.) The profit is about $ "Do not round until the final answer. Then round to the whole number as needed.) 12. An apple farm yields an average of 40 bushels of apples per tree when 24 trees are planted on an acre of ground. Each time 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield? In order to get the highest yield, trees should be planted on an acre. 13. A rectangular box with a volume of 320 ft" is to be constructed with a square base and top. The cost per square foot for the bottom is 20c, for the top is 15e, and for the sides is 2.5p. What dimensions will minimize the cost? What are the dimensions of the box? The length of one side of the base is (1) The height of the box is (2) (Round to one decimal place as needed.) (1) Ofta. (2) Of. Oft. Oft . 14. A retail outlet for calculators sells 720 calculators per year. It costs $2 to store one calculator for a year. To reorder, there is a fixed cost of $5, plus $2.75 for each calculator. How many times per year should the store order calculators, and in what lot size, in order to minimize inventory costs? The store should order calculators times per year to minimize inventory costs. (Simplify your answers.) 15. An open-top cylindrical container is to have a volume 2744 cm. What dimensions (radius and height) will minimize the surface area? The radius of the can is about cm and its height is about cm. 'Do not round until the final answer. Then round to two decimal places as needed.]16. A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 32 ft. What should its dimensions be in order to allow the maximum amount of light to enter through the window? X To allow the maximum amount of light to enter through the window, the radius of the semicircle should be (1) and the height of the rectangle should be (2). (Do not round until the final answer. Then round to two decimal places as needed.) (1) Ofta (2) Oft. Oft2 Oft 2 Oft Oft 3 17. A power line is to be constructed from a power station at point A to an island at point C, which is 5 mi directly out in the water from a point B on the shore. Point B is 6 mi downshore from the power station at A. It costs $4500 per mile to lay the power line under water and $3000 per mile to lay the line under ground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that S could very well be B or A. (Hint: The length of CS is (25+ x2 . ) X 6-x S is miles from A. (Round to two decimal places as needed.) 18. Homing pigeons avoid flying over water. Suppose a homing pigeon is released on an island at point C, which is 13 mi directly out in the water from a point B on shore. Point B is 28 mi downshore from the pigeon's home loft at point A Assume that a pigeon flying over water uses energy at a rate 1.35 times the rate over land. Toward what point $ downshore from A should the pigeon fly in order to minimize the total energy required to get to the home loft at A? X 28-X 28 Total energy = (Energy rate over water) . (Distance over water) + (Energy rate over land) . (Distance over land) Point S is miles away from point A. (Type an integer or decimal rounded to three decimal places as needed.)19. A road is to be built between two cities C, and C, which are on opposite sides of a river of uniform width r. C, is a units from the river and C, is b units from the river, with a s b. A bridge will carry the traffic across the river. Where should the bridge be located in order to minimize the total distance between the cities? Give a general solution using the constants a, p-x b, p, and r as shown in the figure. Bridge-I River P Find D(x), the total distance between the cities. Choose the correct answer below. O A. D(x) = Val + (p-x)2 + vb?+x2 OB. D(X) = Val + (p-x)2 +b2 +x2 +r O c. D(x) = Vaz + (p-x)? +r OD. D(x) = Vb? + x2 +r To find the critical value(s), first find D'(x). Choose the correct answer below. x- p X p - x O A. D'(x) = 162 + x2 OB. D'(x) = Vaz + (p -x)2 b2 + x2 Vaz + (p - x) 2 P - x p - > O c. D'(x) = (b2 - x2 OD. D'(x) = Vaz - (p -x)2 b2 + x2 Vaz + (p - x)2 For what values of x, if any, does D'(x) = 0? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. OA. X= (Use a comma to separate answers as needed.) O B. There are no such values of x. For what values of x, if any, does D'(x) not exist? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. OA. *= (Use a comma to separate answers as needed.) O B. There are no such values of x. The critical value is x = - bp atb . Use the second derivative to determine whether it corresponds to a minimum. bp Consider x = - "atb. Which of the following describes D" 3 +62 OA. D = bp a +b 0 O C. D bp =0 Suppose that f is differentiable for every x in an open interval (a,b) and that there is a critical value c in (a,b) for which f'(c) = 0. Then the Second-Derivative Test says that f(c) is a relative minimum if f"(c) > 0 and f(c) is a relative maximum if f"(c)