Math 241 Homework 6 Solutions

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Naomi Lynne Phelps
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1 Math 241 Homework 6 s Section 3.7 (Pages ) Problem 2. Suppose that the radius r and surface area S = 4πr 2 of a sphere are differentiable functions of t. Write an equation that relates ds/ to /. S = 4πr 2 ds = 8πr Problem 3. The radius r and height h of a right circular cylinder are related to the cylinder s volume V by the formula V = πr 2 h. How is dv / related to dh/ if r is constant? How is dv / related to / if h is constant? (c) How is dv / related to / and dh/ if neither r nor h is constant? If r is a constant then we have V = πr 2 h dv = dh πr2 If h is a constant we have (c) If neither are constants we have V = πr 2 h dv = 2πr h dv = 2πr h + dh πr2 1

2 Problem 12. Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following rates: dx = 1 m/sec, = 2 m/sec, = 1 m/sec. Find the rates at which the box s volume, surface area, and (c) diagonal length s = x 2 + y 2 + z 2 are changing at the instant when x = 4, y = 3, and z = 2. (c) Plugging in dx Plugging in dx Plugging in dx = 1, V = xyz dv = dx yz + x z + xy = 2, = 1, x = 4, y = 3, and z = 2 we have dv = 1(3)(2) + (4)( 2)(2) + (4)(3)(1) = 2m3 /sec S = 2xy + 2xz + 2yz ds = 2x + 2dx y + 2x ds = 1, = 2, + 2dx = 1, x = 4, y = 3, and z = 2 we have z + 2y + 2 z = 2(4)( 2) + 2(1)(3) + 2(4)(2) + 2(1)(2) + 2(3)(1) + 2( 2)(2) = 8 m/sec s = x 2 + y 2 + z 2 ds = 1 2 (x2 + y 2 + z 2 ) 1/2 (2x dx + 2y + 2z ) = 1, = 2, = 1, x = 4, y = 3, and z = 2 we have ds = 1 2 ( ) 1/2 (2(4)(1) + 2(3)(2) + 2(2)(1)) = 1 2 (29) 1/2 (24) = m/s 2

3 Find the rates at which the box s volume, surface area, and (c) diagonal length s = 2x 2 + y 2 + z 2 are changing at the instant when x = 4, y = 3, and z = A sliding ladder A 13-ft ladder is leaning against a house when its base starts to slide away (see accompanying figure). By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft>sec. a. How fast is the top of the ladder sliding down the wall then? Problem 13. A 13-ft ladder is b. leaning At what rate against is the area of the a triangle house formed when by the ladder, its base starts to slide away (see wall, and ground changing then? figure). By the time the base is 12 ft from the house, the base is moving at the y m rate deep? of 5 ft/sec. c. At what rate is the angle u between the ladder and the ground changing then? y y(t) 0 13-ft ladder 24. Commercial air traffic Two commercial airplanes are flying at an altitude of 40,000 ft along straight-line courses that intersect at right angles. Plane A is approaching the intersection point at a speed of 442 knots (nautical miles per hour; a nautical mile is 2000 yd). Plane B is approaching the intersection at 481 knots. At what rate is the distance between the planes changing when A is 5 nautical miles from the intersection point and B is 12 nautical miles from the intersection point? How fast is the top of the ladder sliding down the wall then? 25. Flying a kite A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of 25 ft>sec. How fast must she let out the string when the kite is 500 ft away from her? 26. Boring a cylinder The mechanics at Lincoln Automotive are reboring a 6-in.-deep cylinder to fit a new piston. The machine they are using increases the cylinder s radius one-thousanh of an inch every 3 min. How rapidly is the cylinder volume increasing when the bore (diameter) is in.? u x(t) x Water level y a. At what rate is the water level changing when the water is 8 m deep? b. What is the radius r of the water s surface when the water is c. At what rate is the radius r changing when the water is 8 m deep? 30. A growing rainop Suppose that a op of mist is a perfect sphere and that, through condensation, the op picks up moisture at a rate proportional to its surface area. Show that under these circumstances the op s radius increases at a constant rate. 31. The radius of an inflating balloon A spherical balloon is inflated with helium at the rate of 100p ft 3 >min. How fast is the balloon s radius increasing at the instant the radius is 5 ft? How fast is the surface area increasing? 32. Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft>sec. a. How fast is the boat approaching the dock when 10 ft of rope are out? At what rate is the area of the triangle formed by the ladder, wall, and ground changing then? (c) At what rate is the angle θ between the ladder and the ground changing then? From the pythagorean theorem we have x 2 + y 2 = x dx + 2y = 0 We are given that x = 12 and dx/ = 5. Using the pythagorean theorem we have Plugging these in we have y = = 25 = 5 2(12)(5) + 2(5) = 0 = 120 = 12 ft/sec 10 b. At what rate is the angle u changing at this instant (see the figure)? r 13 Ring at edge of dock u 6' Plugging in the information from we have A = 1 da xy 2 = 1 2 (dx y + x ) da = 1 2 (5(5) + (12)( 12)) = 59.5 ft2 /sec (c) cos θ = x dθ sin θ 13 = 1 dx 13 At the instant we want, we have from the picture that sin θ = 5/13. Plugging in we have 5 dθ 13 = 5 13 dθ = 1 radians/sec 3

4 Problem 18. Water is flowing at the rate of 50 m 3 /min from a shallow concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. How fast (centimeters per minute) is the water level falling when the water is 5 m deep? How fast is the radius of the water s surface changing then? minute. Answer in centimeters per Since there are two similar triangles we have r h = 45 6 r = 45r 6 = 15h 2 V = 1 3 πr2 h = 75πh3 4 dv = 225πh2 dh 4 We are given that dv / = 50 and h = 5. Plugging these in we have 50 = 225π 4 (25)dh dh m/min = 1.13 cm/min r = 15h 2 Plugging in our answer from we have = 15 2 dh 8.49 cm/min Problem 20. Suppose that a op of mist is a perfect sphere and that, through condensation, the op picks up moisture at a rate proportional to its surface area. Show that under these circumstances the op s radius increases at a constant rate. We have Assuming that dv V = 4 3 πr3 dv = 4πr2 = ka, where k is a constant we have 4πr 2 = 4kπr2 = k 4

5 Problem 24. Coffee is aining from 0 a conical filter into x(t) a cylinical coffeepot at the rate of 10 in 3 /min. How fast is the level in the pot rising when the coffee in the cone is 5 in. deep? How fast is the level in the cone falling then? s(t) 34. Making coffee Coffee is aining from a conical filter into a cylinical coffeepot at the rate of 10 in 3 >min. a. How fast is the level in the pot rising when the coffee in the cone is 5 in. deep? b. How fast is the level in the cone falling then? x 36. Moving along a parabola A particle move y = x 2 in the first quaant in such a way t (measured in meters) increases at a stea 10 the angle of inclination u of the line joining origin changing when x = 3 m? 37. Motion in the plane The coordinates of a p xy-plane are differentiable functions of tim -1 m>sec and > =-5 m>sec. How fast tance from the origin changing as it passes throu 38. Videotaping a moving car You are videot stand 132 ft from the track, following a ca 180 mi>h (264 ft>sec), as shown in the ac How fast will your camera angle u be chang right in front of you? A half second later? 6 Camera u 6 How fast is this level falling? 132 Car We are given that dv pot 6 How fast is this level rising? 35. Cardiac output In the late 1860s, Adolf Fick, a professor of physiology in the Faculty of Medicine in Würzberg, Germany, developed one of the methods we use today for measuring how much blood your heart pumps in a minute. Your cardiac output as you read this sentence is probably about 7 L>min. At rest it is likely to be a bit under 6 L>min. If you are a trained marathon runner running a marathon, your cardiac output can be as high as 30 L>min. V Your cardiac pot = πr 2 h = 9πh dv pot = 9π dh output can be calculated with the formula y = Q D, = 10. Plugging this in we have 10 = 9π dh dh = 10 9π in/min 39. A moving shadow A light shines from the high. A ball is opped from the same heigh away from the light. (See accompanying figu shadow of the ball moving along the gro (Assume the ball falls a distance s = 16t 2 ft i 50-ft pole Light 0 30 Ball at time t = 0 1/2 sec later From the picture we see that the radius is always half of the height, i.e. r = h/2 V cone = 1 3 πr2 h = πh3 12 dv cone = π dh 4 h2 We are given that dv cone = 10 and h = 5. Plugging this in we have 10 = 25π 4 dh dh = 8 5π in/min 5

6 Problem 26. A company can manufacture x items at a cost of c(x) thousand dollars, a sales revenue of r(x) thousand dollars, and a profit of p(x) = r(x) c(x) thousand dollars. Find dc/, /, and dp/ for the following values of x and dx/. r(x) = 9x, c(x) = x 3 6x x, and dx/ = 0.1 when x = 2 r(x) = 70x, c(x) = x 3 6x /x, and dx/ = 0.05 when x = 1.5 = 9dx = 9(0.1) = 0.9 dc = dx 3x2 12xdx + 15dx dp = 70dx = 3(2) 2 (0.1) 12(2)(0.1) + 15(0.1) = 0.3 = = 0.6 = 70(0.05) = 3.5 dc = dx 3x2 12xdx 45 dx x 2 dp = 3(1.5) 2 (0.05) 12(1.5)(0.05) 45 (1.5) 2 (0.05) = = 3.5 ( ) =

7 Problem 30. A man 6ft tall walks at the rate of 5 ft/sec towards a streetlight that is 16 ft above the ground. At what rate is the tip of his shadow moving? At what rate is the length of his shadow changing when he is 10ft from the base of the light? Using similar triangles we have that x + y y = 16 6 x = 16 6 y y = 5 3 y dx = 5 = = 3 The distance of the tip is x + y so the rate of the tip is dx + = 5 3 = 8 ft/sec The rate of the length of the shadow is / so it is 3 ft/sec 7

MCV4U1 Worksheet 4.7. dh / dt if neither r nor h is constant?

MCV4U1 Worksheet 4.7. dh / dt if neither r nor h is constant? MCV4U1 Worksheet 4.7 This worksheet serves as an additional exercise to complement the lesson and the examples given. Worksheets may take more than one day to complete. If you are stuck, read again the