Differentiate both sides of the equation with respect to t. dr dv dt (Type an exact answer, using as needed. Type an expression using r as the variable. at the instant the radius is The balloon's radius is increasing at a rate of (Simplify your answer.) Find the related rates equation. dV dh dt dt (Type an exact answer, using it as needed.) A 5.5-foot-tall woman walks at 6 ft/s toward a street light that is 22 ft above the ground. What is the rate of change of the length of her shadow when she is 19 ft from the street light? At what rate is the tip of her shadow moving? Let L be the length of the woman's shadow and let x be the woman's distance from the street light. Write an equation that relates L and x. The bottom of a large theater screen is 6 ft above your eye level and the top of the screen is 20 ft above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of 6 ft/s while looking at the screen. What is the rate of change of the viewing angle 0 when you are 60 ft from the wall on which the screen hangs, assuming the floor is flat (see figure)? Write an equation relating x and 0. 20 ft 6 ft 8 Differentiate both sides of the equation with respect to t. de dx dt When the beam strikes the point Q, its speed along the shore is about (Do not round until the final answer. Then round to the nearest integer as needed.) How does the speed of the beam along the shore vary with the distance between P and Q? A. If Q were closer to P, then the speed of the beam along the shore when it strike Q would be greater. OB. The speed of the beam along the shore is a constant because it depends only on the rate at which the light revolves. OC. The speed of the beam along the shore is a constant because it depends only on the distance from the lighthouse to P. Differentiate both sides of the equation with respect to t. de dt dx dt A lighthouse stands 450 m off a straight shore and the focused beam of its light revolves five times each minute. As shown in the figure, P is the point on shore closest to the lighthouse and Q is a point on the shore 250 m from P. What is the speed of the beam along the shore when it strikes the point Q? Describe how the speed of the beam along the shore varies with the distance between P and Q. Neglect the height of the lighthouse. Q 250 m 450 m Let x be the distance along the shore from point P to the beam and let O be the angle through which the light revolves. Write an equation that relates x and 0. An angler hooks a trout and reels in his line at 3 in./s. Assume the tip of the fishing rod is 11 ft above the water and directly above the angler, and the fish is pulled horizontally directly toward the angler (see figure). Find the horizontal speed of the fish when it is 21 ft from the angler. 11 ft Decreasing at 3 in./s Let x be the horizontal distance from the angler to the fish and z be the length of the fishing line, where both x and z are measured in inches. Write an equation relating x and z. Water is drained out of an inverted cone, having the dimensions depicted in the figure below. If the water level drops at 6 ft/min, at what rate is water (in ft/min) draining from the tank when the water depth is 4 ft? 3 ft 12 ft CH Let V be the volume of water in the tank and let h be the depth of the water. Write an equation that relates V and h. (Type an exact answer, using as needed.) A spherical balloon is inflating with helium at a rate of 72 3 ft min How fast is the balloon's radius increasing at the instant the radius is 3 ft? Write an equation relating the volume of a sphere, V, and the radius of the sphere, r. (Type an exact answer, using as needed.)