Question

the distance between lincoln, ne, and boulder, co, is about 500 miles. the distance between boulder, co, and a third city is 200 miles. assuming the three cities make a triangle on the map, which values represent the possible distance, d, in miles, between lincoln, ne, and the third city? <d<

Explanation:

Step1: Apply the triangle inequality theorem

The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Let \(a = 500\) (distance between Lincoln, NE and Boulder, CO) and \(b=200\) (distance between Boulder, CO and the third - city).

Step2: Calculate the lower and upper bounds

First, calculate the lower bound: \(|500 - 200|=300\).
Then, calculate the upper bound: \(500 + 200 = 700\).

Answer:

\(300\lt d\lt700\)