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: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(|a - b| < c < a + b\). Let \(a = 500\) (distance between Lincoln and Boulder), \(b = 200\) (distance between Boulder and the third city), and \(c = d\) (distance between Lincoln and the third city).

Step2: Apply the theorem

First, calculate the lower bound: \(|500 - 200| = 300\). Then, calculate the upper bound: \(500 + 200 = 700\). So, the inequality for \(d\) is \(300 < d < 700\).

Answer:

\(300 < d < 700\) (assuming the problem is to find the range of possible distances using the triangle inequality theorem, the lower bound is 300 and upper bound is 700)