Question

How fast does the depth of the water change when the wate ris 10 ft hgih if the cone leaks water at a rate of 10 ft3 /minx?

Answer 1

The depth of the water changes at a rate of 10 ft/min when the water is 10 ft high.

Step-by-step explanation:

To determine the rate at which the depth of the water changes when it is 10 ft high, we can use the concept of similar triangles. Let's assume that the depth of the water is x ft at a certain time t. The volume of the water leaking out of the cone in that small time interval is given by the rate of leakage (10 ft³/min) multiplied by the time interval (dt).

Since the cone is leaking water at a constant rate, the volume of water leaving the cone in that time interval is equal to the volume of water that the depth has changed by in that time interval. Therefore, we have:

x/t = 10

To find dx/dt, we differentiate both sides of the equation:

dx/dt = d(10t)/dt = 10.

Therefore, the depth of the water changes at a rate of 10 ft/min when the water is 10 ft high.