Problema Solution
A carpenter is building a rectangular room with a fixed perimeter of 136ft. What dimensions would yield the maximum area? What is the maximum area?
Answer provided by our tutors
Let
l = the length of the rectangular room, l>0
w = the width of the rectangular room, w>0
The perimeter is 136 ft
2(l + w) = 136 divide both sides by 2
l + w = 68
l = 68 - w
The area is calculated by the formula A = l*w.
Plug l = 68 - w into A = l*w
A = (68 - w)*w
A = -w^2 + 68w
We need to find the maximum of the parabolic function A = -w^2 + 68w
A max = c - b^2/(4a), where a = -1, b = 68, c = 0
A max = 0 - 68^2/(4(-1))
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A max = 1,156 ft^2
The maximum area is 1,156 ft^2.
By solving -w^2 + 68w = 1156 we find:
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w = 34 ft
l = 68 - w = 68 - 34 = 34 ft
The dimensions of the rectangular area are: the length is 34 ft and the width is 34 ft.