Question

A person plans to use 90 feet of fencing and the side of his
house to enclose a rectangular garden. What dimensions of
the rectangle would give the maximum area?

Answer 1

so you let the sides of the rectangle be x and y so



so the perimeter is

90 = 2x + y

the area is

A = xy

rewrite the perimeter with y as the subject

90 - 2x = y

now substitute into the area formula

A = x(90 - 2x)

or

A = -2x^2 + 90x

the parabola is concave down so you will have a maximum

find the 1st derivative dA/dx

solve for x

this will give the value of x that gives the maximum area.

to find y substitute x into the perimeter formula .

hope this helps

Answer 2

well if you haven't done calculus use the line of symmerty

the equation is \[A = -2x^2 + 90\]

for the line of symmetry use

\[x = \frac{-b}{2a} \]

b = 90 and a = -2

this the the length, you need to find the width by substituting into the perimeter equation.