1 151 WebCalc Fall 2002-copyright Joe Kahlig Quiz #11 MATH 151 Section Name: October 11, 2002 This is due at the start of class on Thursday. 1. A metal rod has the shape of a right circular cylinder. As it is being heated, its length is increasing at a rate of 0.005cm/min and its diameter is increasing at 0.002cm/min. At what rate is the volume changing when the rod has length 40cm and diameter 3cm? The volume of this rod is given by the formula V = πr 2 l. Since the diameter is given in the problem and d = 2r or d2 = r, we can rewrite the volume as V = π4 d2 l. Given in the problem dl is dt = 0.005, dd dt = .002, l = 40, and d = 3. Now take a derivative and plug in the known values. dV π dd π 2 dl = dl ∗ + d dt 2 dt 4 dt dV π π = ∗ 3 ∗ 40 ∗ .002 + ∗ 9 ∗ .005 dt 2 4 dV = 0.41233 cm/min dt 2. A person flying a kite holds the string 5 ft above the ground level, and the string is released at a rate of 2ft/sec as the kite moves horizontally at an altitude of 105ft (see figure). Assuming there is no sag in the string, find the rate at which the kite is moving when 125 ft of string has been released. Answer: Given in the problem is x = 100, z = 125, dz dx dt = 2, and dt = 0 (since the kite is only moving horizontally). Using the formula x2 + y 2 = z 2 and 105 feet solving for y gives y = 75. Now take a derivative and plug in the known values. 2x dx dt + 2y dy dt = z x y 5 feet 2z dz dt dy 2 ∗ 125 ∗ 2 = = 3.3333f t/sec dt 2 ∗ 75 3. Water is leaking out of an inverted conical tank at a rate of .01m3 /min. The tank has height of 8m and the diameter at the top is 6m. Find the rate of change of the radius (located at the water level of the tank) when the height of the tank is 3m. V = 13 πr 2 h From the problem we know dV dt = −0.01 and h = 3 The other information is labeled on the picture. The relationship between the radius and the height is given by: hr = 83 or h = 83 r V = 13 πr 2 h = 13 πr 2 38 r = 89 πr 3 8 dV 2 dr dr = 3 πr dt note: if h = 3 then r = 2 9 8 −0.01 = π 3 8 Solving for dr dt 9 8 dr dt gives dr dt = −.0009431m/min 6m 3m r 8m h