To find the maximum possible area of the corral given the available 760 feet of fencing, we'll use a rectangular layout where one side is the creek and the other three sides are fenced.
We'll divide the remaining fencing equally for two sides, let's call each of these sides \( x \) feet, and use the remaining fencing for the third side, which we'll call \( 760 - 2x \) feet.
The area \( A \) of the corral is given by the formula:
\[ A = x \times (760 - 2x) \]
To maximize the area, we'll find the derivative of \( A \) with respect to \( x \), set it equal to zero, and solve for \( x \). Then we'll use this value of \( x \) to find the corresponding value of \( A \).
Let's do the math:
\[ A = x(760 - 2x) \]
\[ A = 760x - 2x^2 \]
Now, let's find the derivative of \( A \) with respect to \( x \):
\[ \frac{dA}{dx} = 760 - 4x \]
Setting the derivative equal to zero:
\[ 760 - 4x = 0 \]
\[ 4x = 760 \]
\[ x = \frac{760}{4} \]
\[ x = 190 \]
So, \( x = 190 \) feet.
Now, let's find the corresponding area:
\[ A = 190 \times (760 - 2 \times 190) \]
\[ A = 190 \times (760 - 380) \]
\[ A = 190 \times 380 \]
\[ A = 72,200 \]
Therefore, the maximum possible area of the corral is \( 72,200 \) square feet.
