Answer:
We know that the formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
We are given that the area of the circle is increasing at a constant rate of 352 square feet per second. So we can say that dA/dt = 352 square feet per second.
We want to find the rate of change of the radius at the instant when the radius is 9 feet. We can use the chain rule to relate the rates of change of A and r as follows:
dA/dt = dA/dr * dr/dt
We can rearrange this equation to solve for dr/dt:
dr/dt = (dA/dt) / (dA/dr)
To find dA/dr, we can differentiate the formula for the area of a circle with respect to r:
A = πr^2
dA/dr = 2πr
So dA/dr = 2π(9) = 18π square feet per foot.
Substituting the given values into the formula for dr/dt, we get:
dr/dt = (dA/dt) / (dA/dr)
dr/dt = 352 / (18π)
dr/dt ≈ 6.199 feet per second (rounded to three decimal places)
Therefore, the rate of change of the radius at the instant when the radius is 9 feet is approximately 6.199 feet per second