Question

The length and width of a rectangle must add up to 78 feet. calculate the dimensions that will yield the maximum area (remember, area = length times width). what is the length that results in the maximum area?

Answer 1

The length that results in the maximum area is 39 feet.

Step-by-step explanation

Lets assume, length of the rectangle is
`x` feet and width of the rectangle is
`y` feet.

As the length and width must add up to 78 feet, so the equation will be...

`x+y=78 ................................................(1)`

Solving equation (1) for y :
`y= 78-x`

Now the area of the rectangle,

`A = x*y\\\\ A= x(78-x)\\\\ A= 78x-x^2\\\\ A= -x^2 +78x...................(2)`

`A` will be maximum when
`(dA)/(dx) = 0`

Now taking derivative of equation(2) with respect to
`x`......

`(dA)/(dx)= -2x+78`

If
`(dA)/(dx) =0`, then

`-2x+78=0\\\\ -2x=-78\\\\ x=(-78)/(-2)=39`

If
`x= 39`, then
`y= 78-x = 78-39=39`

So, both length and width will be 39 feet for getting maximum area.

The length that results in the maximum area is 39 feet.