Question

Given a perimeter of 180, how do you find the length and the width of the rectangle of maximum area?

Answer 1

Given a perimeter of 180, the length and width of the rectangle with maximum area are 45 and 45.

Explanation:

Let `x=` the length and `y=` the width of the rectangle.
The area of the rectangle `A =xy`

`2x+2y=180` because the perimeter is `180`.

Solve for `y`
`2y=180-2x`
`y=90-x`

Substitute for `y` in the area equation.
`A=x(90-x)`
`A=90x-x^2`

This equation represents a parabola that opens down. The maximum value of the area is at the vertex.

Rewriting the area equation in the form `ax^2+bx+c`
`A=-x^2+90xcolor(white)(aaa)a=-1, b=90, c=0`

The formula for the `x` coordinate of the vertex is
`x=(-b)/(2a)= (-90)/(2*-1)=45`

The maximum area is found at `x=45`
and `y=90-x=90-45=45`

Given a perimeter of 180, the dimensions of the rectangle with maximum area are 45x45.