Question

The area of a circle is increasing at a constant rate of 352 square feet per second. At the instant when the radius of the circle is 9 feet, what is the rate of change of the radius? Round your answer to three decimal places.

Answer 1

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

Answer 2

The rate of change of the radius when the area of a circle is increasing at a specific rate is found using the derivative of the area formula. At the instant when the radius is 9 feet, the radius is changing at approximately 6.228 feet per second.

Step-by-step explanation:

To find the rate of change of the radius when the area of a circle is increasing at a constant rate, we use the area formula A = πr^2, where A is the area and r is the radius of the circle. Differentiating both sides of the equation with respect to time t gives us dA/dt = 2πr (dr/dt), where dA/dt is the rate of change of the area and dr/dt is the rate of change of the radius.

Given that dA/dt = 352 square feet per second and the radius r = 9 feet at the instant in question, we can solve for dr/dt:

1. dA/dt = 2πr (dr/dt)
2. 352 = 2 × 3.1415927 × 9 × (dr/dt)
3. 
(dr/dt) = 352 / (2 × 3.1415927 × 9)
4. 
(dr/dt) approx 6.228

Therefore, the rate of change of the radius at the instant when the radius is 9 feet is approximately 6.228 feet per second, rounded to three decimal places.