Question

the exercise yard at the corrections center has four sides and is enclosed with $499\frac{7}{24}$ feet of security fencing around it. if three sides of the yard measure $108\frac{2}{3}$ feet, $140\frac{3}{4}$ feet, and $117\frac{7}{8}$ feet, find the length of the fourth side. the length of the fourth side is \\(\square\\) feet (simplify your answer. type a whole number, proper fraction, or mixed number.)

Explanation:

Step1: Convert mixed numbers to improper fractions

First, convert each mixed number to an improper fraction.

• \( 499\frac{7}{24}=\frac{499\times24 + 7}{24}=\frac{11976+7}{24}=\frac{11983}{24} \)
• \( 108\frac{2}{3}=\frac{108\times3 + 2}{3}=\frac{324 + 2}{3}=\frac{326}{3} \)
• \( 140\frac{3}{4}=\frac{140\times4+3}{4}=\frac{560 + 3}{4}=\frac{563}{4} \)
• \( 117\frac{7}{8}=\frac{117\times8 + 7}{8}=\frac{936+7}{8}=\frac{943}{8} \)

Step2: Find a common denominator for the three known sides

The common denominator of 3, 4, and 8 is 24.

• \( \frac{326}{3}=\frac{326\times8}{3\times8}=\frac{2608}{24} \)
• \( \frac{563}{4}=\frac{563\times6}{4\times6}=\frac{3378}{24} \)
• \( \frac{943}{8}=\frac{943\times3}{8\times3}=\frac{2829}{24} \)

Step3: Sum the three known sides

Add the three fractions: \( \frac{2608}{24}+\frac{3378}{24}+\frac{2829}{24}=\frac{2608 + 3378+2829}{24}=\frac{8815}{24} \)

Step4: Subtract the sum from the total perimeter

Let the length of the fourth side be \( x \). We know that the total perimeter \( P = x + \) sum of the three sides. So, \( x=\frac{11983}{24}-\frac{8815}{24}=\frac{11983 - 8815}{24}=\frac{3168}{24} \)

Step5: Simplify the fraction

Simplify \( \frac{3168}{24} \). Divide numerator and denominator by 24: \( \frac{3168\div24}{24\div24}=132 \)

Answer:

132