Question

Among all rectangles that have a perimeter of 182, find the dimensions of the one whose area is largest.

Answer 1

Answer:

The largest rectangle of perimeter 182 is a square of side 45.5

Step-by-step explanation:

Maximization Using Derivatives

The procedure consists in finding an appropriate function that depends on only one variable. Then, the first derivative of the function will be found, equated to 0 and find the maximum or minimum values.

Suppose we have a rectangle of dimensions x and y. The area of that rectangle is:

$A=x.y$

And the perimeter is

$P=2x+2y$

We know the perimeter is 182, thus

$2x+2y=182$

Simplifying

$x+y=91$

Solving for y

$y=91-x$

The area is

$A=x.(91-x)=91x-x^2$

Taking the derivative:

$A'=91-2x$

Equating to 0

$91-2x=0$

Solving

$x=91/2=45.5$

Finding y

$y=91-x=45.5$

The largest rectangle of perimeter 182 is a square of side 45.5