Question

juan is building an irregularly shaped garden that has a perimeter of 15\frac{1}{2}ft. use the picture of his garden below to determine the length of the missing side. include units in your answer. 2\frac{2}{3}ft 4\frac{1}{8}ft 3\frac{3}{4}ft ? ft

Explanation:

Step1: Convert mixed numbers to improper fractions

The perimeter \(P = 15\frac{1}{2}=\frac{15\times2 + 1}{2}=\frac{31}{2}\), \(2\frac{2}{3}=\frac{2\times3+2}{3}=\frac{8}{3}\), \(3\frac{3}{4}=\frac{3\times4 + 3}{4}=\frac{15}{4}\), \(4\frac{1}{8}=\frac{4\times8+1}{8}=\frac{33}{8}\).
Let the missing side be \(x\). The perimeter formula for a polygon (sum of all sides) gives \(P=\frac{8}{3}+\frac{15}{4}+\frac{33}{8}+x\).

Step2: Find a common denominator

The common denominator of \(3\), \(4\), and \(8\) is \(24\).
\(\frac{8}{3}=\frac{8\times8}{3\times8}=\frac{64}{24}\), \(\frac{15}{4}=\frac{15\times6}{4\times6}=\frac{90}{24}\), \(\frac{33}{8}=\frac{33\times3}{8\times3}=\frac{99}{24}\), \(\frac{31}{2}=\frac{31\times12}{2\times12}=\frac{372}{24}\).
The equation becomes \(\frac{372}{24}=\frac{64}{24}+\frac{90}{24}+\frac{99}{24}+x\).

Step3: Simplify the right - hand side

\(\frac{64 + 90+99}{24}+x=\frac{253}{24}+x\).

Step4: Solve for \(x\)

\(x=\frac{372}{24}-\frac{253}{24}=\frac{372 - 253}{24}=\frac{119}{24}=4\frac{23}{24}\)

Answer:

\(4\frac{23}{24}\text{ ft}\)