Questions: An ecologist is conducting a research project on breeding pheasants in captivity. She first must construct suitable pens. She wants a rectangular area with two additional fences across its width, as shown in the sketch. Find the maximum area she can enclose with 936 m of fencing. Select the correct choice below and fill in the answer box(es) to complete your choice. A. The length of the shorter side of the larger rectangular area is 117^7 m and the length of the longer side of the larger rectangular area is 234 m. B. The length of each side of the larger rectangular area is the same and measures m. The largest total area that can be enclosed is

Solution Steps

Given the total length of fencing \(L\) and two additional fences across the width, we have \(2x + 3y = L\). Solving for \(y\), we get \(y = \frac{L - 2x}{3}\).

The area of the rectangle is given by \(A = x \times y\). Substituting \(y\) from Step 1, we get \(A = x \times \frac{L - 2x}{3}\).

The derivative of \(A\) with respect to \(x\) is 312 - 4*x/3, and solving \(\frac{dA}{dx} = 0\) gives the critical points as [234].

The second derivative of \(A\) with respect to \(x\) at the critical point 234 is -4/3. Since it's negative, this critical point corresponds to a maximum area.

The optimal length \(x\) for the maximum area is 2.3E+2, and the corresponding width \(y\) is 1.6E+2.

Final Answer: The maximum area that can be enclosed by the rectangular pen with two additional fences across its width, given a total length of fencing \(L = 936\), is approximately 36504 square meters.