Answer:
781250 [tex]ft^{2}[/tex]
Step-by-step explanation:
Let y is the length of the farm field
Let x is the width of the farm field
Given that, no fencing is necessary along the rock wall, so we can find the perimeter of the farm is:
2x + y = 2500 feet
<=> y = 2500 -2x
The are of the farm has the following formula:
A = x*y
<=> A = x(2500 - 2x)
<=> A = 2500x -2[tex]x^{2}[/tex]
To have the maximum area of field in square feet, we need to use differentials to estimate:
[tex]\frac{dA}{dx}[/tex] = 2500 - 4x
Set [tex]\frac{dA}{dx}[/tex] = 0, we have:
2500 - 4x = 0
<=> x = 625 feet.
=> y = 2500 - 2*625 = 1250 feet
So the maximum area of field is:
A = x*y = 625*1250 = 781250 [tex]ft^{2}[/tex]
