I need help with question #25,29 and 33. I'm not sure how to do it thank you

232 CHAPTER 3 . DERIVATIVES 24. Flying a kite Once Kate's kite reaches a height of 50 ft (above component of velocity is always 30 ft/ s. What is the vertical ponent of velocity when (a) x = 10 and (b) x - 40) her hands), it rises no higher but drifts due east in a wind blowing 5 ft/s. How fast is the string running through Kate's hands at the moment when she has released 120 ft of string? 25. Rope on a boat A rope passing through a capstan on a dock is attached to a boat offshore. The rope is pulled in at a constant rate of 3 ft/s and the capstan is 5 ft vertically above the water. How 400- 30 fu/s fast is the boat traveling when it is 10 ft from the dock? 30 fuls 26. Bug on a parabola A bug is moving along the right side of the 200 parabola y = x at a rate such that its distance from the origin is increasing at 1 cm/min. a. At what rate is the x-coordinate of the bug increasing at the 10 20 30 40 50 point (2, 4)? dx b. Use the equation y = x to find an equation relating to It c. At what rate is the y-coordinate of the bug increasing at the 31. Draining a water heater A water heater that has the shape of a right cylindrical tank with a radius of I ft and a height of 4 ft point (2, 4)? is being drained. How fast is water draining out of the tank (in 27. Balloons and motorcycles A hot-air balloon is 150 ft above fts / min) if the water level is dropping at 6 in/ min? the ground when a motorcycle (traveling in a straight line on a horizontal road) passes directly beneath it going 40 mi/hr 32. Drinking a soda At what rate (in in' / s) is soda being sucked out (58.67 ft/s). If the balloon rises vertically at a rate of 10 ft/s, of a cylindrical glass that is 6 in tall and has a radius of 2 in? The what is the rate of change of the distance between the motorcycle depth of the soda decreases at a constant rate of 0.25 in/s. and the balloon 10 seconds later? 33. Piston compression A piston is seated at the top of a cylindrical 28. Baseball runners Runners stand at first and second base in a chamber with radius 5 cm when it starts moving into the chamber baseball game. At the moment a ball is hit, the runner at first base at a constant speed of 3 cm/s (see figure). What is the rate of runs to second base at 18 ft/s; simultaneously, the runner on sec- change of the volume of the cylinder when the piston is 2 cm from ond runs to third base at 20 ft/s. How fast is the distance between the base of the chamber? the runners changing I s after the ball is hit (see figure)? (Hint: The distance between consecutive bases is 90 ft and the bases lie 5 cm - - Piston at the corners of a square.) 2nd base 90 ft 3rd base Ist base 90 ft Home plate 34. Filling two pools Two cylindrical swimming pools are being filled simultaneously at the same rate (in m' / min; see figure). The smaller pool has a radius of 5 m, and the water level rises at a rate Fishing story An angler hooks a trout and reels in his line at of 0.5 m/ min. The larger pool has a radius of 8 m. How fast is the 4 in/s. Assume the tip of the fishing rod is 12 ft above the water water level rising in the larger pool? and directly above the angler, and the fish is pulled horizontally directly toward the angler (see figure). Find the horizontal speed Inflow rates of the fish when it is 20 ft from the angler. are equal. Decreasing at 4 in/s 12 ft 35. Growing sandpile Sand falls from an overhead bin and accumu lates in a conical pile with a radius that is always three times its It. Suppose the height of the pile increases at a rate of 2 cm/$ when the pile is 12 cm high. At what rate is the sand leaving the Parabolic motion An arrow is shot into the air and moves along bin at that instant? he parabolic path y = x(50 - x) (see figure). The horizo