number 95,100,103

CHAPTER 3 Review 269 (b) Find the rate of change of the volume with respect to the radius if the height is constant. 100. A waterskier skis over the ramp shown in the figure at a speed of 30 ft/s. How fast is she rising as she leaves the 91. The mass of part of a wire is x(1 + Vx) kilograms, where ramp? x is measured in meters from one end of the wire. Find the linear density of the wire when x = 4 m. modity is 92. The cost, in dollars, of producing x units of a certain com- 4 ft - 15 ft C(x) = 920 + 2x - 0.02x2 + 0.00007x3 (a) Find the marginal cost function. 101. The angle of elevation of the sun is decreasing at a rate of (b) Find C'(100) and explain its meaning. 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall (c) Compare C'(100) with the cost of producing the building increasing when the angle of elevation of the sun 101st item. is TT /6? 93. A bacteria culture contains 200 cells initially and grows at a 102. (a) Find the linear approximation to f(x) = V25 - x2 rate proportional to its size. After half an hour the population near 3. has increased to 360 cells. (b) Illustrate part (a) by graphing f and the linear (a) Find the number of bacteria after t hours. approximation. (b) Find the number of bacteria after 4 hours. (c) For what values of x is the linear approximation accurate (c) Find the rate of growth after 4 hours. to within 0.1? d) When will the population reach 10,000? 103. (a) Find the linearization of f(x) = V1 + 3x at a = 0. 94. Cobalt-60 has a half-life of 5.24 years. State the corresponding linear approximation and use it (a) Find the mass that remains from a 100-mg sample after to give an approximate value for V1.03 20 years . (b) Determine the values of x for which the linear approxi- (b) How long would it take for the mass to decay to 1 mg? mation given in part (a) is accurate to within 0.1. 95, Let C(t) be the concentration of a drug in the bloodstream. As 104. Evaluate dy if y = x' - 2x2 + 1, x = 2, and dx = 0.2. the body eliminates the drug, C(t) decreases at a rate that is 105. A window has the shape of a square surmounted by a semi- proportional to the amount of the drug that is present at the circle. The base of the window is measured as having width time. Thus C'(t) = -kC(t), where k is a positive number 60 cm with a possible error in measurement of 0.1 cm. Use called the elimination constant of the drug. differentials to estimate the maximum error possible in (a) If Co is the concentration at time t = 0, find the concen- computing the area of the window. tration at time t. (b) If the body eliminates half the drug in 30 hours, how long 106-108 Express the limit as a derivative and evaluate. does it take to eliminate 90% of the drug? 106. lim - x - 1 uloa 107. lim V16 + h - 2 96. A cup of hot chocolate has temperature 80 C in a room kept x1 x-1 h - 0 h at 20 C. After half an hour the hot chocolate cools to 60 C. Cos 0 - 0.5 (a) What is the temperature of the chocolate after another 108. lim 0- TT/3 0- TT/3 half hour? (b) When will the chocolate have cooled to 40 C? 7. The volume of a cube is increasing at a rate of 10 cm'/min. 1 + tan x - VI + sin x 109. Evaluate lim How fast is the surface area increasing when the length of an edge is 30 cm? 110. Suppose f is a differentiable function such that f(g(x)) = x A paper cup has the shape of a cone with height 10 cm and and f'(x) = 1 + [f(x)]2. Show that g'(x) = 1/(1 + x2). radius 3 cm (at the top). If water is poured into the cup at a 111. Find f'(x) if it is known that rate of 2 cm'/s, how fast is the water level rising when the water is 5 cm deep? " [f ( 2 x ) ] = * 2 A balloon is rising at a constant speed of 5 ft/s. A boy is cycling along a straight road at a speed of 15 ft/s. When he 112. Show that the length of the portion of any tangent line to the passes under the balloon, it is 45 ft above him. How fast is the astroid x2/3 + y2/3 = a2/3 cut off by the coordinate axes is istance between the boy and the balloon increasing 3 s later? constant