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.