Question

A carpenter is building a rectangle room with a fixed perimeter of 172ft. What dimensions would yield the maximum area? What is the maximum area? The length that would yield the maximum area is ? ft

Answer 1

I need the area please?

Answer 2

well, say that the length of the area is x. then the width is (172 - 2x)/2, which is 81 - x

so the area is x(81 - x) = 81x - x^2

to maximize the area, you find the turning point of A = -x^2 + 81x

A = -(x^2 - 81x)

A = -((x - 81/2)^2 - (81)^2/4)

A = -(x - 81/2)^2 + (81)^2/4

so the maximum area is 81*81/4 = 6561/4 square feet

Answer 3

and the length that would yield it is 81/2 feet

Answer 4

1849 is what the answer was

Answer 5

yes, that's because i'm an idiot who thought 172/2 was 86. my lord.

Answer 6

er, was 81***

Answer 7

but it's 86...

so, call the length x, then the width is 86 - x

thus the area is x(86 - x) = -x^2 + 86x

A = -(x^2 - 86x) = -((x - 43)^2 - (43)^2)

so A = -(x - 43)^2 + (43)^2

so maximum area is 43*43 = 1849

and length is 43

Answer 8

width is 43 as well...

it makes sense. the maximum area of a rectangle occurs when it is a square