A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m /min, find the rate at which the water level is rising when the water is 3 m deep. SOLUTION We first sketch the cone and label it as in Figure 3. Let V, r, and & be the volume of the water, the radius of the surface, and the height of the water at time r. where f is measured in minutes. We are given that dV/di =[m'/min and we are asked to find dhar when h is Im. The quantities V and h are related by the equation V =turk but it is very useful to express V as a function of & alone. In order to eliminate r, we use FIGURE 3 the similar triangles in Figure 3 to write and the expression for V becomes V - - 1 3 Now we can differentiate each side with respect to f: dV A dV 50 Substituting h = 3 m and dV/di = 2 m'/min, we have dh The water level is rising at a rate of m/min. What is the value in the red box? What is the value in the blue box? What is the value in the green box? What is the value in the pink box? What is the value in the orange box? What is the value in the brown box? (round to two decimal)