A piston is seated at the top of a cylindrical chamber with radius 3 cm when it starts moving into the chamber at a constant speed of 5 cm/s (see gure). What is the rate of change of the volume of the cylinder when the piston is 3 cm from the base of the chamber? Let V, r, and h be the volume, radius, and height of a cylinder. respectively. Write an equation relating V, r, and h. (Type an exact answer, using 1: as needed.) A rectangle initially has dimensions 5 cm by 6 cm. All sides begin increasing in length at a rate of 4 cm / 5. At what rate is the area of the rectangle increasing after 25 s? Let A, b, and h be the area, base, and height of a rectangle, respectively. Write an equation relating A, b, and h. A 25-foot ladder is leaning against a vertical wall (see gure) when Jack begins pulling the foot of the ladder away from the wall at a rate of DJ M 5. How fast is the top of the ladder sliding down the wall when the foot of the ladder is 24 ft from the wall? Let x be the distance from the foot of the ladder to the wall and let y be the distance from the top of the ladder to the ground. Write an equation relating x and y. Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always two times its height. Suppose the height of the pile increases at a rate of 2 cm / s when the pile is 19 cm high. At what rate is the sand leaving lhe bin at that instant? Let V and h be the volume and height of the cone, respectively. Write an equation that relates V and h and does not include the radius of the cone. (Type an exact answer, using 1: as needed) An inverted conical water tank with a height of 6 ft and a radius of 3 fl is drained through a hole in the vertex at a rate of 6 ft3 / 5 (see gure). What is the rate of change of the water depth when the water depth is 4 ft? (Hint: Use similar triangles.) 6G I I :Outow 6 zls 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 hi (Type an exact answer, using 1: as needed.) Water is drained out of an inverted cone, having the dimensions depicted in the gure below. If the water level drops at 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 he 8 ft/ min, at what rate is water (in a / min) draining from the tank when the water depth is 2 ft? '7 41? (Type an exact answer, using 1|: as needed.) A hotair balloon is 180 ft above the ground when a motorcycle (traveling in a straight line on a horizontal road) passes directly beneath it going 30 mi / hr (44 ft/ 5). If the balloon rises vertically at a rate of 15 it] 5, what is the rate of change of the distance between the motorcycle and the balloon 9 seconds later? Let x be the horizontal distance from the balloon to the motorcycle, y be the vertical distance from the balloon to the road. and 1 be the distance between the motorcycle and the balloon. Write an equation relating x, y, and z
