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Chapter 4 | Applications of Derivatives
31. A tank is shaped like an upside-down square pyramid, with base of 4 m by 4 m and a height of 12 m (see the following figure). How fast does the height increase when the water is 2 m deep if water is being pumped in at a rate of 2 3 m/sec?
39. A lighthouse, L , is on an island 4 mi away from the closest point, P , on the beach (see the following image). If the lighthouse light rotates clockwise at a constant rate of 10 revolutions/min, how fast does the beam of light move across the beach 2 mi away from the closest point on the beach?
For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 ft 3 /min. The radius of the pool is 10 ft. 32. Find the rate at which the depth of the water is changing when the water has a depth of 5 ft. 33. Find the rate at which the depth of the water is changing when the water has a depth of 1 ft. 34. If the height is increasing at a rate of 1 in./sec when the depth of the water is 2 ft, find the rate at which water is being pumped in. 35. Gravel is being unloaded from a truck and falls into a pile shaped like a cone at a rate of 10 ft 3 /min. The radius of the cone base is three times the height of the cone. Find the rate at which the height of the gravel changes when the pile has a height of 5 ft. 36. Using a similar setup from the preceding problem, find the rate at which the gravel is being unloaded if the pile is 5 ft high and the height is increasing at a rate of 4 in./min. For the following exercises, draw the situations and solve the related-rate problems. 37. You are stationary on the ground and are watching a bird fly horizontally at a rate of 10 m/sec. The bird is located 40 m above your head. How fast does the angle of elevation change when the horizontal distance between you and the bird is 9 m? 38. You stand 40 ft from a bottle rocket on the ground and watch as it takes off vertically into the air at a rate of 20 ft/ sec. Find the rate at which the angle of elevation changes when the rocket is 30 ft in the air.
40. Using the same setup as the previous problem, determine at what rate the beam of light moves across the beach 1 mi away from the closest point on the beach. 41. You are walking to a bus stop at a right-angle corner. You move north at a rate of 2 m/sec and are 20 m south of the intersection. The bus travels west at a rate of 10 m/ sec away from the intersection – you have missed the bus! What is the rate at which the angle between you and the bus is changing when you are 20 m south of the intersection and the bus is 10 m west of the intersection? For the following exercises, refer to the figure of baseball diamond, which has sides of 90 ft.
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