Example 2 (expressing rates of change in terms of particular variables). (a) Express the rate of change of the surface area S, of a rectangular solid in terms of the width w, length 1, height h, and the rates of change of w, I, and h. (b) Express the rate of change of the volume, V, of a sphere in terms of the radius r, and the rate of change of the radius. (c) Express the rate of change of the volume V, of a cylinder in terms of the radius r, height h. and rates of change of r and h. Example 3: (a) ) A spherical beach ball is being pumped so that its volume is increasing at a rate of 120 cubic centimeters per second. How fast is the radius of the balloon increasing when the diameter is 30 cm? (b) Assuming the same situation as part (a), how fast is the surface area of the beach ball increasing when the diameter is 40 cm? Example 4: Some Guy (his name) walks along a straight path at a rate of 5 fsee. What hasn't been said yet is that Some Guy just escaped from prison. He is very nonchalant, and doesn't feel the need to run. Before he realizes it, a searchlight at ground level has spotted him. The searchlight is located 25 ft from the path and is kept focused on Some Guy. At what rate is the searchlight rotating when Some Guy is 20 feet from the point on the path closest to the searchlight? Example 5 A woman owns a house that has a strangely shaped swimming pool. The swimming pool is in the shape of an inverted circular cone with radius 8 feet and depth of 16 feet. The pool had recently been emptied for extensive cleaning and is now ready to be filled again. Water is pumped into the pool at a rate of 8 cubic feet per second. How fast is the water level rising when the depth of the water is 9 feet?l