Answer:
The largest area that can be enclosed is 9248 square feet.
Step-by-step explanation:
This is a typical problem of optimization that can be solved using derivatives. We have a rectangular region, and let us denote the height by , and the base by .
Then, the area of the rectangle is . Notice that the area is a function of and , but if we want to use calculus, we should have only one variable. This can be done if we find a relationship between both variables.
Recall that the fences will not bu used in the whole perimeter of the rectangular area, but only in three sides. Hence, . (Without lost of generality we can consider , instead.)
Then, ans substituting in the formula for the area:
Taking derivative with respect to x:
Its only zero can be found solving the equation . Hence, its only zero of is . In order to assure that 68 is a point of maximum, we find and conclude that, in effect, 68 is a point of maximum.
We obtain the value of substituting the value of in the relationship between both variables: . With the values of and we can calculate the desired area: