Question

The radius of a cone is increasing at a constant rate of 4 feet per second. The volume remains a constant 446 cubic feet. At the instant when the radius of the cone is 9 feet, what is the rate of change of the height? The volume of a cone can be found with the equation V = ⅓πr²h. Round your answer to three decimal places.

Answer 1

The rate of change of the height when the radius of the cone is 9 feet is approximately -8.745 feet per second.

Step-by-step explanation:

To find the rate of change of the height when the radius of the cone is 9 feet, we can differentiate the volume equation with respect to time. The equation is V = 1/3 * πr²h. Since the volume remains constant at 446 cubic feet, we have 446 = 1/3 * π(9)²h. Solving for h, we get h = 446 / (1/3 * π(9)²), which is approximately 2.90992 feet. Hence, the rate of change of the height is the derivative of h with respect to time, which is dh/dt.

To differentiate h with respect to t, we can use implicit differentiation. Differentiating both sides of the equation h = 446 / (1/3 * π(9)²), we get dh/dt = -17944/729π, which is approximately -8.745 feet per second. Therefore, the rate of change of the height is approximately -8.745 feet per second.

Answer 2

To find the rate of change of height, differentiate the volume equation with respect to time and substitute the given values. The rate of change of height, dh/dt, is -2/9 feet per second.

To find the rate of change of height, we can use the formula for the volume of a cone, which is V = ⅓πr²h. We are given that the volume remains constant at 446 cubic feet. At the instant when the radius, r, is 9 feet, we need to find the rate of change of height, dh/dt. We can solve for dh/dt by differentiating the volume equation with respect to time and substituting the given values.

First, we differentiate the volume equation: dV/dt = ⅓π(2rhdr/dt + r²dh/dt).

Since the volume is constant, dV/dt = 0. Plugging in the given values, we have 0 = ⅓π(2(9)(4) + 9²dh/dt).

Simplifying the equation, we get 0 = ⅓π(18 + 81dh/dt).

So, we have 0 = 6π + 27πdh/dt. Solving for dh/dt, we get dh/dt = -6/27 = -2/9 feet per second.