Certainly! To determine the possible distance [tex]\(d\)[/tex] between Lincoln, NE, and the third city, given the distances between Lincoln and Boulder, and Boulder and the third city, we can use the triangle inequality theorem. Here’s how it works step-by-step:
1. Understand the Given Distances:
- The distance between Lincoln, NE, and Boulder, CO: [tex]\(500\)[/tex] miles
- The distance between Boulder, CO, and the third city: [tex]\(200\)[/tex] miles
2. Apply the Triangle Inequality Theorem:
The triangle inequality theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Hence, for three sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] of a triangle, the following must hold true:
- [tex]\(a + b > c\)[/tex]
- [tex]\(a + c > b\)[/tex]
- [tex]\(b + c > a\)[/tex]
3. Setting Up the Inequalities:
Let:
- [tex]\(a\)[/tex] be the distance from Lincoln to Boulder ([tex]\(500\)[/tex] miles)
- [tex]\(b\)[/tex] be the distance from Boulder to the third city ([tex]\(200\)[/tex] miles)
- [tex]\(c\)[/tex] be the distance from Lincoln to the third city ([tex]\(d\)[/tex] miles)
We need to find the bounds for [tex]\(d\)[/tex]. According to the triangle inequality theorem:
- [tex]\(500 + 200 > d\)[/tex]
- [tex]\(500 + d > 200\)[/tex]
- [tex]\(200 + d > 500\)[/tex]
4. Simplify the Inequalities:
- From [tex]\(500 + 200 > d\)[/tex]:
[tex]\[
d < 700
\][/tex]
- From [tex]\(500 + d > 200\)[/tex]:
[tex]\[
d > -300
\][/tex]
However, distances cannot be negative, so we disregard the negative value.
- From [tex]\(200 + d > 500\)[/tex]:
[tex]\[
d > 300
\][/tex]
5. Determine the Possible Range for [tex]\(d\)[/tex]:
Combining the valid inequalities, we get:
[tex]\[
300 < d < 700
\][/tex]
Therefore, the possible distance [tex]\(d\)[/tex], in miles, between Lincoln, NE, and the third city, given the distances from Lincoln to Boulder and from Boulder to the third city, ranges between [tex]\(300\)[/tex] and [tex]\(700\)[/tex] miles.
So the final answer is:
[tex]\[ 300 < d < 700 \][/tex]
