Question

The perimeter of the patio shown below is 120 feet. Develop a formula to find y in terms of x. Explain your work.

Answer 1

Perimeter of the given patio is
`(\pi+6)y + 10x +8`.

Explanation:

Let us first label the given patio using the points A, B, C, D, E and F as shown the attachment diagram.

Perimeter of given figure can be found by the following:

Perimeter of polygon ABPCDEQF + Perimeter of semi-circle with diameter as EF.

Let us mark points P and Q on sides AB and FA respectively.

These points P and Q are just opposite to point D. Please refer to attached diagram for details.

As this figure has symmetry so, sides

`side\ AP =side\ FE\ i.e\ 2y\\side\ AQ =side\ BC\ i.e.\ 2y\\side\ PB =side\ CD\ i.e.\ 3x+5\\side\ QF =side\ DE\ i.e.\ 2x-1`

Perimeter of polygon APBCDEQF = AP + PB + BC + CD + DE + FQ + QA (Side EF is not on the periphery so it is excluded)

`\Rightarrow 2y + (3x+5) + 2y + (3x+5) + (2x - 1) + (2x-1) + 2y\\\Rightarrow 6y + 10x + 8 ...... (1)`

Perimeter of semicircle =
`\pi * radius`

Radius of semicircle is y.

`\text{Perimeter of semicircle = }\pi * y ...... (2)`

Adding equations (1) and (2):

`\text{Total perimeter = }6y + 10x + 8 + \pi y \\\Rightarrow (\pi+6)y + 10x + 8`

Perimeter of the given figure is
`(\pi+6)y + 10x +8`