Question

The radius of a cone is increasing at a constant rate of 4 meters per second, and the volume is increasing at a rate of 904 cubic meters per second. at the instant when the radius of the cone is 1010 meters and the volume is 489 cubic meters, what is the rate of change of the height?

Answer 1

The rate of change of the height of the cone is approximately 0.067 meters per second.

Step-by-step explanation:

To find the rate of change of the height, we need to use the formula for the volume of a cone and differentiate it with respect to time.

The formula for the volume of a cone is: V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height.

We are given that the radius is increasing at a rate of 4 meters per second (dr/dt = 4 m/s) and the volume is increasing at a rate of 904 cubic meters per second (dV/dt = 904 m³/s).

At the instant when the radius is 1010 meters and the volume is 489 cubic meters, we can substitute these values into the volume formula.

Solving for the rate of change of the height (dh/dt), we can differentiate the volume formula with respect to time and solve for dh/dt.

dh/dt = (3dV/dt) / (4πr²)

Substituting the given values, we have:

dh/dt = (3 * 904) / (4 * π * (1010)²)

dh/dt ≈ 0.067 meters per second

Answer 2

To find the rate of change of the height of the cone, the volume formula for a cone is differentiated with respect to time and known rates are substituted to solve for dh/dt.

Step-by-step explanation:

The problem involves finding the rate of change of the height (dh/dt) of a cone, given that the radius is increasing at 4 meters per second (dr/dt = 4 m/s) and the volume is increasing at 904 cubic meters per second (dV/dt = 904 m³/s). To find the rate of change of the height, we'll use the formula for the volume of a cone, V = (1/3) π r² h, and apply the Chain Rule to this expression regarding time.

Given:
dr/dt = 4 m/s
dV/dt = 904 m³/s
r = 10 m
V = 489 m³

Let's differentiate the volume formula with respect to time (t):
dV/dt = (1/3)π(2r·(dr/dt)·h + r²·(dh/dt))

Substitute the known values and solve for dh/dt:
904 = (1/3)π(2·10·4·h + 10²·(dh/dt))

After calculating h from the volume formula and then substituting all the values: dh/dt = specific value to be calculated m/s