Question

Farmer Ed has 950 meters of​ fencing, and wants to enclose a rectangular plot that borders on a river. If Farmer Ed does not fence the side along the​ river, find the length and width of the plot that will maximize the area. What is the largest area that can be​ enclosed?

Answer 1

The perimeter for this case is given by:

$P = y + 2x$

Substituting values we have:

$y + 2x = 950$

The area is given by:

$A = x * y$

Writing the area based on a variable we have:

$A (x) = x * (950 - 2x)\\A (x) = -2x ^ 2 + 950x$

We derive the area to obtain the maximum of the function:

$A '(x) = -4x + 950$

We equal zero and clear x:

$-4x + 950 = 0\\4x = 950\\x = 950/4\\x = 237.5$

Then, the other dimension is given by:

$y = 950 - 2x\\y = 950 - 2 * (237.5)\\y = 475$

Finally the maximum area is:

$A = (237.5) * (475)\\A = 112812.5 m ^ 2$

Answer:

The length and width of the plot that will maximize the area are:

$x = 237.5 m\\y = 475 m$

The largest area that can be enclosed is:

$A = 112812.5 m ^ 2$