In a triangle, the lengths of any two sides must be greater than the length of the third side, and the sum of the lengths of any two sides must be greater than the length of the third side.
Given sides of lengths 15 ft and 22 ft, let's denote the length of the third side as \( x \).
We can derive the following inequalities based on the triangle inequality theorem:
- \( 15 + 22 > x \)
- \( 15 + x > 22 \)
- \( 22 + x > 15 \)
Now, let's solve each inequality:
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From \( 15 + 22 > x \): \[ 37 > x \quad \text{or} \quad x < 37 \]
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From \( 15 + x > 22 \): \[ x > 22 - 15 \quad \Rightarrow \quad x > 7 \]
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From \( 22 + x > 15 \): \[ x > 15 - 22 \quad \Rightarrow \quad x > -7 \quad \text{(this is always true since x is a side length)} \]
Combining the relevant inequalities (first and second), we find: \[ 7 < x < 37 \]
Thus, the range of values for the third side \( x \) is: \[ \boxed{(7, 37)} \]