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

Answer:

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

Step-by-step 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 = $\pi \times radius$

Radius of semicircle is y.  

$\text{Perimeter of semicircle = }\pi \times y ...... (2)$

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 $(\pi+6)y + 10x +8$