To determine which length of fencing could complete Jerald's triangular pen, we need to consider the properties of a triangle. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side (Triangle Inequality Theorem).
Since Jerald has rolls of wire that are 15 ft and 22 ft long, we can denote the sides of the triangle as \( a = 15 \) ft, \( b = 22 \) ft, and \( c \) being the unknown length of fencing that could be used to complete the triangle.
To satisfy the triangle inequality, the following conditions must hold:
- \( a + b > c \) (15 + 22 > c)
- \( a + c > b \) (15 + c > 22)
- \( b + c > a \) (22 + c > 15)
Let's solve these inequalities step by step.
-
From the first inequality, \( 15 + 22 > c \) leads to: \[ 37 > c \quad \text{or} \quad c < 37 \]
-
The second inequality, \( 15 + c > 22 \) leads to: \[ c > 22 - 15 \] \[ c > 7 \]
-
The third inequality, \( 22 + c > 15 \) leads to: \[ c > 15 - 22 \quad \text{which simplifies to} \quad c > -7 \] This condition is always true since \( c > 7 \) is a stronger requirement.
Now we combine these results: \[ 7 < c < 37 \]
Now, we can evaluate the given options:
- Option #1: 35 ft. – This value is within the range (7, 37).
- Option #2: 40 ft. – This value is outside the range (greater than 37).
- Option #3: 42 ft. – This value is outside the range (greater than 37).
- Option #4: 45 ft. – This value is also outside the range (greater than 37).
The only option that fits within the valid range for the length of fencing \( c \) is Option #1: 35 ft.