Question: An inverted cone with height 10 cm and radius 2 cm is partially filled with a mysterious green liquid, which is oozing through the sides of the cone at a rate proportional to the surface area of the cone in contact with the liquid. (The surface area of a cone is given by , where is the radius of the cone and is the "slant height" of the cone.) The green liquid is being replenished by being poured into the top of the cone by aliens at a rate of 1 cm3/min. At this rate, when the depth of the fluid is 4 cm, the depth is decreasing at the rate of 0.1 cm/min. The Aliens want to keep the depth of the fluid constant at 4 cm. By how much should they change the rate at which they are replenishing the fluid in the cone?
Hint: Find two expressions for the rate of change of volume. The first one comes from the formula for the volume of a cone, and the second comes from the difference between the fluid flowing in and the fluid oozing out. Use this to solve for your proportionality constant (traditionally called k), then go from there.