Question

A fence for a rectangular garden with one side against an existing wall is constructed by using 60 feet of fencing. What is the maximum area that can be enclosed

Answer 1

Step-by-step explanation:

A fence for a rectangular garden with one side against an existing wall is constructed by using 60 feet of fencing.

Perimeter of rectangle (3 sides)= 60 feet

Let 'x' be the width of the wall

$Perimeter = 2(length)+2(width)\\60=2(length)+2x\\\\60-2x=2(length)\\\frac{60-2x}{2} =length\\Length =30-x$

Formula for the area of the rectangle is

$Area=length \cdot width\\A=length(x)$

Replace the length we got using perimeter

$A=(30-x)(x)\\A(x)= 30x-x^2$

To find out the maximum are we take derivative

$A'(x)= 30-2x\\0=30-2x\\-30=-2x\\x=15$

find out second derivative to check whether x=15 is maximum

$A''(x)=-2$

second derivative is negative

So, Maximum area at x=15

To find maximum area we plug in 15 for x in A(x)

$A(x)=30x-x^2\\A(15)=30(15)-15^2\\A(15)=225$

So, maximum area is 225 square feet