Answer:
[tex]A = 43.74[/tex]
Step-by-step explanation:
Represent Length with L, Width with W and Perimeter with P
[tex]P = 27[/tex]
[tex]L=x+3[/tex]
[tex]W = \frac{2}{3}(x + 3)[/tex]
Required
Determine the Area
From the perimeter, we need to solve for x;
[tex]P = 2(L + W)[/tex]
Substitute values for P, L and W
[tex]27 = 2(x + 3 + \frac{2}{3}(x+3))[/tex]
[tex]27 = 2(x + 3 + \frac{2}{3}x+2)[/tex]
Collect Like Terms
[tex]27 = 2(3 + 2 + x + \frac{2}{3}x)[/tex]
[tex]27 = 2(5 + \frac{3x + 2x}{3})[/tex]
[tex]27 = 2(5 + \frac{5x}{3})[/tex]
Open bracket
[tex]27 = 10 + \frac{10x}{3}[/tex]
Collect Like Terms
[tex]27 - 10 = \frac{10x}{3}[/tex]
[tex]17= \frac{10x}{3}[/tex]
Solve for x
[tex]x = \frac{3*17}{10}[/tex]
[tex]x = \frac{51}{10}[/tex]
[tex]x = 5.1[/tex]
The Area is then calculated using:
[tex]A = L * W[/tex]
[tex]A = (x + 3)*\frac{2}{3}(x+3)[/tex]
[tex]A = \frac{2}{3}(x+3)^2[/tex]
Substitute 5.1 for x
[tex]A = \frac{2}{3}(5.1+3)^2[/tex]
[tex]A = \frac{2}{3}(8.1)^2[/tex]
[tex]A = \frac{2}{3}*65.61[/tex]
[tex]A = \frac{2*65.61}{3}[/tex]
[tex]A = \frac{131.22}{3}[/tex]
[tex]A = 43.74[/tex]
Hence, the area is [tex]43.71cm^2[/tex]
