Find the rate of change of the area of a circle with respect to the radius r. What is the rate when r = 12

A = pi r^2

dA/dr = 2 pi r which is the circumference of course so DA = circumference * dr

if r = 12
2 pi r = 24 pi

π cm2/sec

To find the rate of change of the area of a circle with respect to the radius r, we can use the formula for the area of a circle.

The formula for the area of a circle is A = πr^2, where A represents the area and r represents the radius.

To find the rate of change with respect to the radius, we need to differentiate the equation with respect to r.

dA/dr = d(πr^2)/dr

Using the power rule of differentiation, we can differentiate r^2 as 2r:

dA/dr = 2πr

Now, we can substitute the given value of r = 12 into the equation to find the rate of change of the area when r = 12:

dA/dr = 2π(12)
= 24π

Therefore, the rate of change of the area of a circle with respect to the radius when r = 12 is 24π.

To find the rate of change of the area of a circle with respect to the radius, we can use the derivative. The area formula of a circle is given by A = πr^2, where A represents the area and r represents the radius.

To find the derivative of the area with respect to the radius, we differentiate the area formula with respect to r.

dA/dr = d(πr^2)/dr

Using the power rule of differentiation where d(x^n)/dx = nx^(n-1), we can rewrite the derivative as:

dA/dr = 2πr

Now, to find the rate of change of the area when r = 12, we substitute the value of r into the equation:

dA/dr = 2π(12)

Simplifying, we get:

dA/dr = 24π

Therefore, the rate of change of the area of a circle with respect to the radius when r = 12 is 24π.