If Jerald is enclosing a triangular area with fencing for a chicken pen and he has rolls of wire that are 15 ft. long and 22 ft. long, which of the following pieces of fencing could he use to complete his pen?

Question

Option #1: 35 ft.
Option #2: 40 ft.
Option #3: 42 ft.
Option #4: 45 ft.

Answer 1

To determine which of the given options could potentially complete Jerald's triangular pen, we need to consider the properties of a triangle. Specifically, the lengths of the three sides must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.

Jerald has rolls of wire measuring 15 ft. and 22 ft. Let's denote these lengths as:

• \( a = 15 \) ft
• \( b = 22 \) ft
• \( c \) = the length of the fencing to complete the triangle (one of the options)

To explore the possible options, we can denote the options as follows:

1. \( c_1 = 35 \) ft
2. \( c_2 = 40 \) ft
3. \( c_3 = 42 \) ft
4. \( c_4 = 45 \) ft

Now, we can analyze each option to see if they can form a valid triangle with the other two sides.

For \( c_1 = 35 \) ft:

• Check the triangle inequalities:
  1. \( 15 + 22 > 35\) → \( 37 > 35\) (True)
  2. \( 15 + 35 > 22\) → \( 50 > 22\) (True)
  3. \( 22 + 35 > 15\) → \( 57 > 15\) (True)

All inequalities are satisfied. So, 35 ft is a valid option.

For \( c_2 = 40 \) ft:

• Check the triangle inequalities:
  1. \( 15 + 22 > 40\) → \( 37 > 40\) (False)
  2. \( 15 + 40 > 22\) → \( 55 > 22\) (True)
  3. \( 22 + 40 > 15\) → \( 62 > 15\) (True)

One inequality is false. So, 40 ft is not a valid option.

For \( c_3 = 42 \) ft:

• Check the triangle inequalities:
  1. \( 15 + 22 > 42\) → \( 37 > 42\) (False)
  2. \( 15 + 42 > 22\) → \( 57 > 22\) (True)
  3. \( 22 + 42 > 15\) → \( 64 > 15\) (True)

One inequality is false. So, 42 ft is not a valid option.

For \( c_4 = 45 \) ft:

• Check the triangle inequalities:
  1. \( 15 + 22 > 45\) → \( 37 > 45\) (False)
  2. \( 15 + 45 > 22\) → \( 60 > 22\) (True)
  3. \( 22 + 45 > 15\) → \( 67 > 15\) (True)

One inequality is false. So, 45 ft is not a valid option.

Conclusion:

Jerald can use 35 ft of fencing to complete his triangular pen.