Question

Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 20 feet of fencing to go around it, what dimensions will give him the maximum area in the garden?

Answer 1

You need to find the rectangle area and semicircle area. Then add them together.

Answer 2

Here is how to find the maximum area of a rectangle with a fixed perimeter. In short it turns out to be a square. so the relationship you need to solve is the semicircle length to the other three equal sides.

Perimeter = 2(L + W)
100 = 2(L+W)
L+W = 50
---
Let width be W
Then length = 50-W
--------

Area = length * width
A = (50-W)W
A = 50W-W^2
--
This is a quadratic with a = -1, b = 50:
Maximum area occurs when W = -b/2a = -50/-2 = 25
Since W = 25 and L+W=50, L = 25

Answer 3

@ehuman are you sure thats right
if you demonstrate it in a drawing i will tick you as best response