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
[tex]Perimeter = 2(length)+2(width)\\60=2(length)+2x\\\\60-2x=2(length)\\\frac{60-2x}{2} =length\\Length =30-x[/tex]
Formula for the area of the rectangle is
[tex]Area=length \cdot width\\A=length(x)[/tex]
Replace the length we got using perimeter
[tex]A=(30-x)(x)\\A(x)= 30x-x^2[/tex]
To find out the maximum are we take derivative
[tex]A'(x)= 30-2x\\0=30-2x\\-30=-2x\\x=15[/tex]
find out second derivative to check whether x=15 is maximum
[tex]A''(x)=-2[/tex]
second derivative is negative
So, Maximum area at x=15
To find maximum area we plug in 15 for x in A(x)
[tex]A(x)=30x-x^2\\A(15)=30(15)-15^2\\A(15)=225[/tex]
So, maximum area is 225 square feet
