To determine the possible distance \(d\) between Lincoln, NE, and the third city, assuming the three cities make a triangle on the map, we can use the triangle inequality theorem. This theorem states that, for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Given:
- Distance between Lincoln, NE, and Boulder, CO: \(500\) miles
- Distance between Boulder, CO, and the third city: \(200\) miles
Let \(d\) be the distance between Lincoln, NE, and the third city.
According to the triangle inequality theorem, we have the following inequalities:
1. \(d\) + \(200\) miles > \(500\) miles
2. \(d\) + \(500\) miles > \(200\) miles
3. \(500\) miles + \(200\) miles > \(d\)
By solving these inequalities:
1. \(d + 200 > 500\)
- Subtract \(200\) from both sides: \(d > 300\)
2. \(d + 500 > 200\)
- Subtract \(500\) from both sides: \(d > -300\) (which is always true as \(d\) is a positive distance)
3. \(500 + 200 > d\)
- Simplify: \(700 > d\)
Combining these inequalities, we find the possible range for \(d\):
[tex]\[ 300 < d < 700 \][/tex]
So, the possible distance \(d\) between Lincoln, NE, and the third city must satisfy:
[tex]\[ 300 < d < 700 \][/tex]
