how do you find the area of this?

Question

A gardener wishes to enclose her rectangular raspberry patch with exactly 40 feet of fencing. The raspberry bushes are planted along the garage, so no fencing is needed on that side. 

a. Find the dimensions if the total area is to be 192 square feet. 

Answer 1

Let L1, L2 and L3 be the 3 sides requiring fencing. L1+L2+L3 = 40.

Let L2 = L3, so the above equation can be rewritten as L1 + 2L2 = 40 (1)

L1*L2 = 192 = area of the enclosed plot. (2)

Solve (1) for L1:

L1 = 40 - 2L2.

Plug this into (2) to get (40 - 2L2) * L2 = 192 ==> 40L2 - 2L2² = 192. (3)

Rearrange terms of (3) to get 2L2² - 40L2 + 192 = 0

Divide both sides by 2 to get L2² - 20L2 + 96 = 0

Use the quardratic equation to find 2 solutions for L2 = 12 and 8

When L2 = 12, L1 = 16. Since L2 = L3, L1+L2+L3 = 16+12+12 = 40.

12 x 16 = 192.

12 and 16 satisfy conditions for length of fencing and area.

Now try L2 = 8. L2 = L3, so L1 = 40 - 2*8 = 24

24 x 8 = 192.

24 and 8 satisfy conditions for length of fencing and area.

Thus there are 2 possible solutions to the problem.