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The problem provided involves calculating the expected area of a circular sampling region, where the radius \(R\) is a random variable with a given probability density function (pdf) \(f(r)\).

### PDF of the radius \(R\):
\[
f(r) = 
\begin{cases} 
\frac{3}{4} \left[1 - (11 - r)^2 \right], & 10 \leq r \leq 12 \\
0, & \text{otherwise}
\end{cases}
\]

We are asked to find the expected area of the circular region. The area \(A\) of a circle is related to the radius by \(A = \pi r^2\). Therefore, we need to calculate the expected value of \(A\), which is:

\[
\mathbb{E}[A] = \mathbb{E}[\pi R^2] = \pi \mathbb{E}[R^2]
\]

Now, to compute \(\mathbb{E}[R^2]\), we use the formula for the expected value:

\[
\mathbb{E}[R^2] = \int_{10}^{12} r^2 f(r) \, dr
\]

Substituting \(f(r)\) into this integral, we get:

\[
\mathbb{E}[R^2] = \int_{10}^{12} r^2 \cdot \frac{3}{4} \left[ 1 - (11 - r)^2 \right] \, dr
\]

Next, I'll calculate this integral step by step. 

### Steps to calculate \(\mathbb{E}[R^2]\):
1. Expand the term \(\left[ 1 - (11 - r)^2 \right]\).
2. Multiply by \(r^2\) and integrate from 10 to 12.
3. Multiply the result by \(\pi\) to find \(\mathbb{E}[A]\).

Let me proceed with the calculation.The expected area of the resulting circular region is approximately \( 380.76 \, \text{m}^2 \) when rounded to two decimal places.

Would you like more details on the steps or have any questions? Here are some related questions to explore:

1. How does the shape of the probability density function (pdf) affect the expected value of the radius?
2. What is the variance of the area of the circle given this distribution?
3. How would the expected area change if the range of \(r\) were expanded or contracted?
4. What is the probability that the radius \(r\) falls between 10.5 and 11.5 meters?
5. How would the expected area change if the pdf were shifted to be centered around a different value?

**Tip:** To calculate expected values for functions of random variables, always multiply the function by the probability density function and integrate over the appropriate range.

An ecologist wishes to mark off a circular sampling region having radius 11 m. However, the radius of the resulting region is actually a random variable R with the given pdf. What is the expected area of the resulting circular region?

Learn how to compute the expected area of a circular sampling region with a random radius, given by a probability density function (pdf). This detailed solution covers essential concepts in probability theory, expected value, and integral calculus. Suitable for advanced high school or undergraduate students.

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