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

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:

And the perimeter is

We know the perimeter is 182, thus

Simplifying

Solving for y

The area is

Taking the derivative:

Equating to 0

Solving

Finding y

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

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step-by-step explanation: