Question

parks the developer of the city park shown plans to put a decorative fence around the park. the budget allows for at least 399 feet of fencing but no more than 798 feet to be used. image of a right triangle with legs labeled x and x + 30, hypotenuse labeled x + 90 which could be the length of the side of the park labeled x + 30?

Explanation:

Step1: Find the perimeter formula

The park is a triangle with sides \( x \), \( x + 30 \), and \( x + 90 \). The perimeter \( P \) is the sum of the sides: \( P = x + (x + 30) + (x + 90) = 3x + 120 \).

Step2: Set up the inequality for the perimeter

The budget allows for at least 399 feet and at most 798 feet of fencing. So, \( 399 \leq 3x + 120 \leq 798 \).

Step3: Solve the left inequality

Subtract 120 from all parts: \( 399 - 120 \leq 3x + 120 - 120 \leq 798 - 120 \), which simplifies to \( 279 \leq 3x \leq 678 \).

Step4: Solve for \( x \)

Divide all parts by 3: \( \frac{279}{3} \leq \frac{3x}{3} \leq \frac{678}{3} \), so \( 93 \leq x \leq 226 \).

Step5: Find the range for \( x + 30 \)

Add 30 to all parts of the inequality for \( x \): \( 93 + 30 \leq x + 30 \leq 226 + 30 \), which gives \( 123 \leq x + 30 \leq 256 \).

So, any length between 123 feet and 256 feet (inclusive) could be the length of the side labeled \( x + 30 \). For example, if we take a value like 150 (which is between 123 and 256), that would be a valid length.

Answer:

A length between 123 feet and 256 feet (e.g., 150 feet, or any value in \([123, 256]\))