A circular oil spill on the surface of the ocean spreads outward. Find the approximate rate of change in the area of the oil slick with respect to its radius when the radius is 100... A circular oil spill on the surface of the ocean spreads outward. Find the approximate rate of change in the area of the oil slick with respect to its radius when the radius is 100 m.
Understand the Problem
The question is asking us to find the rate at which the area of a circular oil spill changes as its radius increases, specifically when the radius is 100 m. To solve this, we will use the formula for the area of a circle and differentiate it with respect to the radius.
Answer
The rate at which the area changes is approximately $628.32 \, m^2/m$.
Answer for screen readers
The rate at which the area of the circular oil spill changes when the radius is 100 m is approximately $628.32 , m^2/m$.
Steps to Solve
- Formula for the area of a circle
The area $A$ of a circle is given by the formula:
$$ A = \pi r^2 $$
where $r$ is the radius of the circle.
- Differentiate the area with respect to the radius
To find the rate at which the area changes as the radius increases, we need to find the derivative of the area function with respect to $r$:
$$ \frac{dA}{dr} = \frac{d}{dr} (\pi r^2) $$
Using the power rule of differentiation, we get:
$$ \frac{dA}{dr} = 2\pi r $$
- Substitute the given radius
Now, substitute $r = 100$ m into the derivative to find the rate of change of the area when the radius is 100 m:
$$ \frac{dA}{dr} \bigg|_{r=100} = 2\pi(100) $$
This simplifies to:
$$ \frac{dA}{dr} \bigg|_{r=100} = 200\pi $$
- Calculate the numerical value
Finally, calculate the approximate numerical value:
$$ 200\pi \approx 200 \times 3.14 \approx 628.32 $$
So, we can conclude the area is changing at a rate of approximately 628.32 $m^2$ per meter increase in radius.
The rate at which the area of the circular oil spill changes when the radius is 100 m is approximately $628.32 , m^2/m$.
More Information
The formula for the area of a circle and its derivative are foundational concepts in calculus, specifically in related rates problems. Understanding how changes in one variable (like the radius) affect another (like the area) is crucial in various fields such as environmental science and physics.
Tips
- Forgetting to square the radius in the area formula can lead to incorrect derivatives.
- Misapplying differentiation rules; make sure to apply the power rule correctly.
- Not substituting the radius value correctly into the derivative can lead to wrong rates of change.
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