SOLUTION: a carpenter is building a rectangular room with a fixed perimeter of 128 ft. what dimensions would yield the maximum area? what is the maximum area?

Question 402284: a carpenter is building a rectangular room with a fixed perimeter of 128 ft. what dimensions would yield the maximum area? what is the maximum area?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Let's solve this one in general, that is for any given perimeter.

Let P represent the fixed perimeter.

Let w represent the width of the rectangle.

Let l represent the length of the rectangle.

The perimeter of a rectangle is:

The area of a rectangle is the length times the width so a function for the area in terms of the width is:

Algebra Solution:

The area function is a parabola, opening downward, with vertex at:

Since the parabola opens downward, the vertex represents a maximum value of the area function. The value of the width that gives this maximum value is one-fourth of the given perimeter. Therefore, the shape must be a square, and the area is the width squared.

Calculus Solution:

The area function is continuous and twice differentiable over its domain, therefore there will be a local extrema wherever the first derivative is equal to zero and that extreme point will be a maximum if the second derivative is negative at that point.