To determine the possible values for the third side of a triangle when two sides are given as 7 inches and 11 inches, we need to apply the triangle inequality theorem. The theorem states:
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
2. Mathematically, for sides \(\(a\)\), \(\(b\)\), and \(\(c\)\) of a triangle:
- \(\(a + b > c\)\)
- \(\(a + c > b\)\)
- \(\(b + c > a\)\)
Let \(\(a = 7\)\), \(\(b = 11\)\), and \(\(c\)\) be the third side.
First, we consider:
\(\[ a + b > c \]\)
\(\[ 7 + 11 > c \]\)
\(\[ 18 > c \]\)
So, \(\(c\)\) must be less than 18.
Next, we consider the absolute difference between the lengths of the two sides to ensure it establishes a lower boundary:
\(\[ |a - b| < c \]\)
\(\[ |7 - 11| < c \]\)
\(\[ 4 < c \]\)
So, \(\(c\)\) must be greater than 4.
Combining both conditions, we get:
\(\[ 4 < c < 18 \]\)
Since \(\(c\)\) is specified to be an integer, the possible integer values for \(\(c\)\) are:
\(\[ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 \]\)
From these values, the smallest possible value for \(\(c\)\) is 5, and the largest possible value for \(\(c\)\) is 17.
The product of the smallest and largest values of \(\(c\)\) is:
\(\[ 5 \times 17 = 85 \]\)
Thus, the product of the smallest and largest values that the third side could be is \(\( \boxed{85} \)\).
