Question

Find the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 116 meters

Answer 1

Length = 29 m

Width = 29 m

Explanation:

Let x and y be the length and width of the rectangle, respectively.

The area and perimeter are given by:

`A=xy\\p=116=2x+2y\\y=58-x`

Rewriting the area as a function of x:

`A(x) = x(58-x)\\A(x) = 58x-x^2`

The value of x for which the derivate of the area function is zero, is the length that maximizes the area:

`A(x) = 58x-x^2\\(dA)/(dx)=0=58-2x\\ x=29\ m`

The value of y is:

`y = 58-29\\y=29\ m`

Length = 29 m

Width = 29 m