SOLUTION: A carpenter is building a rectangular room with a fixed perimeter of 332 ft. What dimensions would yield the maximum area? What is the maximum area? The length that would yield the
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Question 328142: A carpenter is building a rectangular room with a fixed perimeter of 332 ft. What dimensions would yield the maximum area? What is the maximum area? The length that would yield the maximum area is how many feet? Answer by solver91311(24713) (Show Source):
Let's solve this one in general, that is for any given perimeter.
Let w represent the width of the field.
Let l represent the length of the field.
The perimeter of a rectangle is:
So
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 available fencing. Therefore, the shape must be a square, and the area is the width squared.
Calculus Solution:
The area function is continuous and twice differentiable across 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.
