Question

1. tess is designing a garden in the shape of a triangle. she knows that she wants one side to have a length of 20 feet and another side to have a length of 14 feet. she would like place fencing along each side of the garden. if she has 60 feet of fencing, is she guaranteed to have enough to surround the garden? justify.
2. if tess wants the third side of the garden to have an integer length, what is the smallest amount of fencing she would need?

Explanation:

Step1: Determine the range of the third side length

According to the triangle - inequality theorem, for a triangle with side lengths \(a = 14\), \(b = 20\), and \(c\) (the third side), \(|a - b|\lt c\lt a + b\).
So, \(|14 - 20|\lt c\lt14 + 20\), which simplifies to \(6\lt c\lt34\).

Step2: Calculate the perimeter for the smallest integer value of \(c\)

Since \(c\) is an integer and \(c>6\), the smallest integer value of \(c\) is \(7\).
The perimeter \(P=a + b + c\). Substitute \(a = 14\), \(b = 20\), and \(c = 7\) into the formula: \(P=14 + 20+7\).
\(P=41\)

Answer:

The smallest amount of fencing she would need is \(41\) feet.