di 14. Radius A spherical balloon is inflated with gas at the rate dx (b) - when x = 25 ay = 2 of 800 cubic centimeters per minute. dt (a) Find the rates of change of the radius when r = 30 centimeters 4. y = 3x2 - 5x (a) dy dt - when x = 3 dx = 2 and r = 85 centimeters. di (b) Explain why the rate of change of the radius of the sphere is dx when x = 2 dy _ 4 not constant even though dV/di is constant. (b) It dt 15. Volume All edges of a cube are expanding at a rate of 5. xy = 4 dy 6 centimeters per second. How fast is the volume changing (a)- - when x = 8 dx - 10 dt when each edge is (a) 2 centimeters and (b) 10 centimeters? dx dy (b) It when x = 1 = -6 16. Surface Area All edges of a cube are expanding at a dt rate of 6 centimeters per second. How fast is the surface dy dx area changing when each edge is (a) 2 centimeters and 6. x2 + y2 = 25 = 8 (a) di when x = 3, y = 4 dt (b) 10 centimeters? dx dy 17. Height At a sand and gravel plant, sand is falling off a (b) - -2 dt - when x = 4, y = 3 dt conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately Moving Point In Exercises 7-10, a point is three times the altitude. At what rate is the height of the pile moving along the graph of the given function at changing when the pile is 15 feet high? (Hint: The formula for the rate dx/dt. Find dy/dt for the given values of x. the volume of a cone is V = 3arch.) 18. Height The volume of oil in a cylindrical container is 7. y = 2x2 + 1; dt dx - 2 centimeters per second increasing at a rate of 150 cubic inches per second. The height of the cylinder is approximately ten times the radius. At what (a) r= -1 (b) x = 0 (c) x = 1 rate is the height of the oil changing when the oil is 35 inches dx high? (Hint: The formula for the volume of a cylinder is 8. y = 1 1 + x2' di = 6 inches per second V = arch.) (a) x = -2 (b) x = 0 (c) x = 2 19. Depth A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the 9. y = tan x; dt dx = 3 feet per second deep end (see figure). Water is being pumped into the pool at cubic meter per minute, and there is 1 meter of water at the (a) x = - 3 (b) x = A (c) x =0 deep end. 1 m3 dx = 4 centimeters per second 4 min 10. y = cos x, dt (a) x= (b) x = = (C) x = 6 m