SOLUTION: A farmer has 114 feet of fencing. Which shape would give the most area? (a) Equilateral triangle (b) Circle (c) Square (d) Non-square rectangle

Question 1209550: A farmer has 114 feet of fencing. Which shape would give the most area?
(a) Equilateral triangle
(b) Circle
(c) Square
(d) Non-square rectangle
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

For a fixed perimeter, the square will always beat the non-square rectangle in terms of larger area.
See this page for a proof.

We can rule out choice (d) since choice (c) has a larger area when the perimeter is fixed.
It could be possible that (c) is the answer, but of course we need to compare with choices (a) and (b).

To do such a comparison, we need the area of the square.
Given a fixed perimeter P, the square's side length is 0.25P and area is (0.25P)^2
The square will have area (0.25P)^2 = (0.25*114)^2 = 812.25 square feet.
This area value is exact and hasn't been rounded.
Highlight this value since we'll come back to it later.

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P = 114 is the perimeter of the equilateral triangle.
Each side is x = P/3 = 114/3 = 38 feet long.

area = 0.25x^2*sqrt(3)
area = 0.25*38^2*sqrt(3)
area = 625.270342 square feet (approximate)
Highlight this value since we'll come back to it later.

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r = radius
C = circumference = distance around the circle's edge = circle's perimeter
C = 2*pi*r
r = C/(2*pi)
r = 114/(2*pi)

A = area of the circle
A = pi*r^2
A = pi*( C/(2*pi) )^2
A = (C^2)/(4pi)
A = (114^2)/(4pi)
A = 1034.18882 square feet is the approximate area of the circle.
I used my calculator's stored version of pi to get the most accuracy possible.

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Let's size up the areas of each shape

equilateral triangle = 625.270342 square feet (approx)
square = 812.25 square feet (exact)
circle = 1034.18882 square feet (approx)

The areas are sorted from smallest to largest.

The circle has the largest area for a fixed perimeter, i.e. fixed amount of fencing.
The farmer should go with the circle. The next best shape is a square.

Answer: Choice (b)

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