The area of a circle is increasing at a constant rate of 178 square feet per second. At the instant when the radius of the circle is 44 feet, what is the rate of change of the radius? Round your answer to three decimal places.

We are asked to find the rate of change of the radius of a circle when the area of the circle is increasing at a constant rate.

### Step 1: Relationship Between Area and Radius
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
where \( r \) is the radius.

### Step 2: Differentiate with Respect to Time
We want to relate the rate of change of the area \( \frac{dA}{dt} \) to the rate of change of the radius \( \frac{dr}{dt} \). To do this, we'll differentiate both sides of the area equation with respect to time \( t \):
\[
\frac{dA}{dt} = 2\pi r \frac{dr}{dt}
\]
Here, \( \frac{dA}{dt} \) is the rate of change of the area, and \( \frac{dr}{dt} \) is the rate of change of the radius.

### Step 3: Plug in the Known Values
We are given:
- \( \frac{dA}{dt} = 178 \) square feet per second (rate of change of area),
- \( r = 44 \) feet (the radius at the instant we're interested in).

Substitute these values into the differentiated equation:
\[
178 = 2\pi (44) \frac{dr}{dt}
\]

### Step 4: Solve for \( \frac{dr}{dt} \)
Now, solve for \( \frac{dr}{dt} \):
\[
\frac{dr}{dt} = \frac{178}{2\pi(44)}
\]
\[
\frac{dr}{dt} = \frac{178}{88\pi}
\]
\[
\frac{dr}{dt} \approx \frac{178}{276.46} \approx 0.644 \text{ feet per second}
\]

Thus, the rate of change of the radius is approximately **0.644 feet per second**.

Would you like more details or explanations on any part of this solution?

### Related Questions:
1. How would the result change if the area increased at a different constant rate?
2. What happens to the rate of change of the radius as the radius increases further?
3. How does the formula for the circumference of a circle relate to this problem?
4. How can we

Learn how to find the rate of change of a circle's radius when its area is increasing at a constant rate of 178 square feet per second. This problem uses calculus and related rates to solve for dr/dt when the radius is 44 feet. Suitable for AP Calculus students.

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