Question

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

Answer 1

Answer:

Length = 29 m

Width = 29 m

Step-by-step 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:

$A(x) = 58x-x^2\\\frac{dA}{dx}=0=58-2x\\ x=29\ m$

The value of y is:

$y = 58-29\\y=29\ m$

Length = 29 m

Width = 29 m