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This problem will prepare you for the Concept Connection on page 364 The solid lines in the figure show an airline's routes between four cities.

a. A traveler wants to fly from Jackson $U$ ) to Shelby $(S)$ but there is no direct flight between these cities. Given that $\mathrm{m} \angle N S J<\mathrm{m} \angle H S I,$ should the traveler first fly to Newton Springs ( $N$ ) or to Hollis ( $H$ ) if he wants to minimize the number of miles flown? Why?

b. The distance from Shelby ( $S$ ) to Jackson $U$ ) is 182 mi. What is the minimum number of miles the traveler will have to fly?

a. Newton Springs

b. 418 miles

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So this real world application is about taking a flight from Jackson to Shelby. However, a direct flight from Jackson's Shelby does not exist. You can either go through Newton Springs or Hollis, and the first thing you want to find out is which flight would be the shortest flight. And also, what is the minimum miles you would have to fly to get there. So a how do we determine what is the shortest flight? Well, we've been given the information that the measurement of angle and SJ is a smaller angle. Then the measurement of angle S H S J. We have the flight from Jackson to Shelby is 100 and 82 that's gonna be the same length in each of these triangles. Um, we also have the distance from Newton Springs to Shelby at the distance from Hollister. Shelby is the same distance of 300 miles. So by hinge there, um, if the included angle in between those measurements is not congruent, we can determine that the distance from Jackson to Newton Springs is gonna be a shorter distance. Then the distance from, uh, Jackson to Hollis. So a we want to take the flight from Jackson to Newton Springs to Shelby is going to be our shortest flight part B. We have to determine what is the minimum number of miles. So I want to focus on the left triangle. We have two out of three sides of that triangle. We learned the theorem that stated we could find all possible measures of the third side by our triangle inequality serum. And it states the sum of any two sides must be greater than the third side. So let's assume that the side we're looking for is the shorter side of the three sides of the triangle. Or at least it's shorter than 300 miles. If it was longer than 300 miles, it wouldn't be a shorter distance. So we could say the current 300 miles plus the distance from Jackson to Show B, which is 182 miles, has to be greater than our distance from Jackson to Newton Springs. Well, that's not really gonna help us find what that measurement is. So we've been any quality that states if you subtract your two numbers, so 300 minus 182 is going to be less than our distance, which we know is also less than 300 plus 1 82 So the minimum distance is going to be our 300 minus 1 82 which is 118. That's gonna be the shortest possible distance from Jackson to Newton Newton Springs. So if we add that to the distance from Jackson to Newton Springs plus Newton Springs to Shelby, we're going to get our minimum distance. So that gives us 300. I'm sorry, our first instances 1 18 plus the current 300 that's going to give us 418 miles. This is the minimum distance you would have to fly from Jackson to Shelby.