at the instant that the first circle has a radius of 9 centimeters, the radius is increasing at a rate of 3 2 centimeters per second. find the rate at which the area of the circle is changing at that instant. indicate units of measure.

The derivative can be thought of as a change rate.
If two variables are related, the derivative of the initial variable to respect to the second variable can be used to calculate the rate of change of one of the variation with respect to the other variable.

For the given question;

Express the circle's area as a function of the its radius.

A = πr²

Determine the area's derivative with respect towards its radius.

dA/dr = 2πr

Using the chain rule, calculate the derivative of a area with respect to time.

dA/dt = dA/dr . dr/dt

dA/dt = 2πr(9)

dA/dt = 18πr

When the radius is 32 cm, find the derivative of a area with respect to time.

dA/dt = 18π(32)

dA/dt = 576π

As a result, the area of a circle is changing at a rate of 576π cm²/sec when the radius of the circle is 32 cm.

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