SOLUTION: Is it possible to have a rectangular property which has perimeter of 78 m and an area of 395m^2?

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Question 1197235: Is it possible to have a rectangular property which has perimeter of 78 m and an area of 395m^2?
Found 3 solutions by Alan3354, math_tutor2020, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Is it possible to have a rectangular property which has perimeter of 78 m and an area of 395m^2?
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Max area of a rectangle with a given perimeter is of a square.
---> 19.5^2 = 1521/4 = 380.25 sq meters max.
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A circle with a circumference of 78 m has an area of ~484 sq meters.

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

L = length
W = width

P = perimeter of a rectangle
P = 2(L+W) = 2L+2W

Replace P with the stated perimeter 78 and let's isolate L
P = 2L+2W
78 = 2L+2W
78/2 = 2L/2+2W/2
39 = L+W
L = 39-W

A = area of a rectangle
A = LW
A = (39-W)W
A = -W^2+39W

Plug in the stated area A = 395
I'll replace each W with x
A = -W^2+39W
395 = -x^2+39x
395+x^2-39x = 0
x^2-39x+395 = 0

Use the quadratic formula to solve for x.
Use a = 1, b = -39, c = 395.
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-39%29%2B-sqrt%28%28-39%29%5E2-4%281%29%28395%29%29%29%2F%282%281%29%29

x+=+%2839%2B-sqrt%281521-1580%29%29%2F%282%29

x+=+%2839%2B-sqrt%28-59%29%29%2F%282%29
The discriminant d+=+b%5E2+-+4ac+=+%28-39%29%5E2-4%281%29%28395%29+=+-59 is negative, so there are no real number solutions.
The two roots of x^2-39x+395 = 0 are complex numbers in the form a+bi where i+=+sqrt%28-1%29
If your teacher hasn't covered complex or imaginary numbers just yet, then ignore this subsection.
The summary is that x^2-39x+395 = 0 has no real solutions.

Since x^2-39x+395 = 0 has no real solutions, this means we cannot have a rectangle with perimeter 78 meters and area 395 square meters.

Answer by ikleyn(52785) About Me  (Show Source):
You can put this solution on YOUR website!
.

A rectangle with a given perimeter which has the maximal area is a square.

For the proof and to see many associated solved problems, look into the lesson
    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
in this site.