Please provide explanations for 27, 31, 32, 33

an inch every 3 min. How rapidly is the cylinder volume increase ing when the bore (diameter) is 3.800 in.? 27. A growing sand pile Sand falls from a conveyor belt at the rate of 10 m/ min onto the top of a conical pile. The height of the pile 33. A balloon and a bicycle A balloon is rising vertically above a is always three-eighths of the base diameter. How fast are the (a) level, straight road at a constant rate of 1 ft/sec. Just when the height and (b) radius changing when the pile is 4 m high? Answer balloon is 65 ft above the ground, a bicycle moving at a constant in centimeters per minute. rate of 17 ft/sec passes under it. How fast is the distance s(1) 28. A draining conical reservoir Water is flowing at the rate of between the bicycle and balloon increasing 3 sec later 50 m'/ min from a shallow concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. a. How fast (centimeters per minute) is the water level falling when the water is 5 m deep? b. How fast is the radius of the water's surface changing then? Answer in centimeters per minute. y(1) 29. A draining hemispherical reservoir Water is flowing at the rate of 6 m'/ min from a reservoir shaped like a hemispherical bowl of radius 13 m, Shown here in profile. Answer the following ques- tions, given that the volume of water in a hemispherical bowl of radius R is V = (7 /3)y?(3R - y) when the water is y meters deep. s ( 1 ) Center of sphere 13 Water level x(1) 34. Making coffee Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 in'/min. a. How fast is the level in the pot rising when the coffee in the a. At what rate is the water level changing when the water is 8 m cone is 5 in. deep? deep? fast is the level in the cone falling then? b. What is the radius r of the water's surface when the water is y m deep? c. At what rate is the radius r changing when the when the water is 8 m deep? 30. A growing raindrop Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these How fast is this circumstances the drop's radius increases at a constant rate. level falling? 31. The radius of an inflating balloon A spherical balloon is inflated with helium at the rate of 1007 ft / 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. How fast The rope is hauled in at the rate of 2 ft/sec. is this a. How fast is the boat approaching the dock when 10 ft of rope level rising? are out ? MacBook Pro