Verified Answer :
- Identify the area formula \(A = \pi r^2\) and the given rate \(\frac{dr}{dt} = 5\) ft/min.
- Differentiate the area formula with respect to time to get \(\frac{dA}{dt} = 2\pi r \frac{dr}{dt}\).
- Substitute \(r = 40\) and \(\frac{dr}{dt} = 5\) into the derivative equation.
- Determine the final rate of change: \(\frac{dA}{dt} = 400\pi \approx \boxed{1256.64}\) square feet per minute.
Explanation
- Problem Analysis
The problem involves a circular oil spill that is expanding over time. We are given the rate at which the radius of the circle is increasing, which is \(\frac{dr}{dt} = 5\) feet per minute. We need to determine how quickly the area of the circle is changing at the specific moment when the radius \(r\) is \(40\) feet. The relationship between the area \(A\) and the radius \(r\) is defined by the formula \(A = \pi r^2\).
- Differentiating the Area Equation
To find the rate of change of the area with respect to time, we need to differentiate the area formula \(A = \pi r^2\) with respect to time \(t\). Applying the chain rule, we have:
\[\frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = 2\pi r \frac{dr}{dt}\]
This derivative gives us the general expression for the rate of change of the area of the circle in terms of its current radius and the rate of change of that radius.
- Calculating the Rate of Change
Now, we substitute the known values into our expression for \(\frac{dA}{dt}\). We are given that the radius \(r = 40\) feet and the rate of change of the radius \(\frac{dr}{dt} = 5\) feet per minute. Substituting these into the formula:
\[\frac{dA}{dt} = 2 \cdot \pi \cdot 40 \text{ ft} \cdot 5 \text{ ft/min}\]
Multiplying these constants and variables together:
\[\frac{dA}{dt} = 400\pi \text{ ft}^2\text{/min}\]
To get a numerical approximation, we use \(\pi \approx 3.14159\):
\[\frac{dA}{dt} \approx 400 \cdot 3.14159 \approx 1256.64 \text{ ft}^2\text{/min}\]
The area is increasing at this rate at the exact moment when the radius is 40 feet.
- Final Answer
The rate at which the area of the circular spill is changing when the radius is 40 feet is \(\boxed{400\pi}\) square feet per minute, which is approximately \(\boxed{1256.64}\) square feet per minute.
Examples
In environmental science, professionals often monitor oil spills to manage their containment. By calculating the rate at which the surface area grows, responders can predict how fast the spill will reach sensitive areas like coastlines. This allows for timely deployment of containment booms and cleanup crews, demonstrating how calculus helps protect our ecosystems.