14.1 EXERCISES 1. In Example 2 we considered the function W-f(T, v), where W is the wind-chill index, T is the actual temperature, and vis the wind speed. A numerical representation is given in Table 1 on page 889, - (a) What is the value of f(-15, 40)? What is its meaning? (b) Describe in words the meaning of the question "For what value of v is f(-20, v) -30?" Then answer the question. (c) Describe in words the meaning of the question "For what value of T is f(7, 20)=-49?" Then answer the question. (d) What is the meaning of the function W-f(-5, v)? 2. The temperature-humidity index I (or humidex, for short) is the perceived air temperature when the actual temperature is 7 and the relative humidity is h, so we can write I= f(T, h). The fol- lowing table of values of I is an excerpt from a table compiled by the National Oceanic & Atmospheric Administration. d. F Describe the behavior of this function. (e) What is the meaning of the function W-f(7,50)? Describe the behavior of this function. emperature Actual T h 20 Table 3 Apparent temperature as a function 80 77 85 82 87 95 93 90 100 T 99 H of temperature and humidity E + Relative humidity (%) 30 78 84 90 96 104 40 79 86 93 101 110 50 81 88 96 60 82 107 90 100 114 120 132 70 83 93 106 124 144 (a) What is the value of f(95, 70)? What is its meaning? (b) For what value of h is f(90, h) 100? (c) For what value of T is f(T, 50) = 88? (d) What are the meanings of the functions I = f(80, h) and I-f(100, h)? Compare the behavior of these two functions of h. 3. A manufacturer has modeled its yearly production function P (the monetary value of its entire production in millions of dollars) as a Cobb-Douglas function P(L, K) = 1.47L0.65 0.35 where L is the number of labor hours (in thousands) and Kis the invested capital (in millions of dollars). Find P(120, 20) and interpret it. 4. Verify for the Cobb-Douglas production function P(L, K)=1.01L0.75 K 0.25 SECTION 14.1 Functions of Several Variables discussed in Example 3 that the production will be doubled if both the amount of labor and the amount of capital are doubled. Determine whether this is also true for the general production function P(L, K)=bL"K¹- 5. A model for the surface area of a human body is given by the function Wind speed (knots) where w is the weight (in pounds), h is the height (in inches), and S is measured in square feet. (a) Find f(160, 70) and interpret it. (b) What is your own surface area? 6. The wind-chill index W discussed in Example 2 has been modeled by the following function: W(T, v) V Check to see how closely this model agrees with the values in Table 1 for a few values of T and v. 7. The wave heights h in the open sea depend on the speed var of the wind and the length of time that the wind has been blowing at that speed. Values of the function h = f(v, t) are recorded in feet in Table 4. 10 15 20 S-f(w, h) 0.109104250.725 - (a) What is the value of f(40, 15)? What is its meaning? (b) What is the meaning of the function h = f(30, t)? Describe the behavior of this function. (c) What is the meaning of the function h = f(v, 30)? Describe the behavior of this function. 1 M 30 40 50 60 13.12 +0.62157-11.37p16+0.3965T016 ++ 5 2 4 5 9 14 19 24 +++ 10 2 4 7 13 21 29 37 Table 4 Duration (hours) 15 2 5 8 16 25 36 47 20 2 5 8 17 28 40 54 30 2 5 9 18 31 45 62 899 ▬▬▬▬ 40 2 5 9 19 33 48 67 50 2 5 T 9 19 33 50 69 8. A company makes three sizes of cardboard boxes: small, medium, and large. It costs $2.50 to make a small box,