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) (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) (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.
The discriminant 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
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(52786) (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.
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