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.

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 $\frac{dA}{dx} = 0$

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

$\frac{dA}{dx}= -2x+78$

If $\frac{dA}{dx} =0$, then

$-2x+78=0\\\\ -2x=-78\\\\ x=\frac{-78}{-2}=39$

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.