Question

using the paths shown, how long is the shortest route from springfield to lexington? springfield 14.4 mi stafford brookfield 7.6 mi 7.5 mi 9 mi 4.5 mi belmont lexington
mi submit

Explanation:

Step1: Calculate the length of the lower path from Springfield to Stafford.

The lower path from Springfield to Stafford goes through Belmont, so we add the distances: \(7.6 + 7.5 = 15.1\) miles. The direct path from Springfield to Stafford is \(14.4\) miles. Since \(14.4 < 15.1\), we take the direct path from Springfield to Stafford (\(14.4\) miles) as the shorter segment from Springfield to Stafford.

Step2: Calculate the total length of the shortest route from Springfield to Lexington.

Now, from Stafford to Lexington, the distance is \(9\) miles (we ignore the path through Brookfield as we are going to Lexington, not Brookfield). So we add the distance from Springfield to Stafford (\(14.4\) miles) and from Stafford to Lexington (\(9\) miles): \(14.4+9 = 23.4\) miles? Wait, no, wait. Wait, maybe I made a mistake. Wait, let's re - examine the graph. Wait, the path from Springfield to Stafford: two options, direct (\(14.4\)) or through Belmont (\(7.6 + 7.5=15.1\)). So Springfield to Stafford is \(14.4\). Then Stafford to Lexington: the path is \(9\) miles (since Lexington is on that path). Wait, but wait, is there a shorter path? Wait, no, let's check again. Wait, maybe I misread the graph. Wait, the nodes: Springfield, Belmont, Stafford, then Stafford to a node, then to Lexington? Wait, no, the graph: Springfield connected to Stafford (14.4) and to Belmont (7.6). Belmont connected to Stafford (7.5). Then Stafford connected to a node (let's say Node X) with 9 miles, and Node X connected to Lexington? Wait, no, the label says "Lexington" is the node after the 9 - mile path? Wait, the original problem: "Using the paths shown, how long is the shortest route from Springfield to Lexington?" Let's list all possible routes:

Route 1: Springfield -> Stafford (14.4) -> Lexington (9). Total: \(14.4 + 9=23.4\)

Route 2: Springfield -> Belmont (7.6) -> Stafford (7.5) -> Lexington (9). Total: \(7.6+7.5 + 9=24.1\)

Wait, but there is another path? Wait, the node after Stafford with 9 miles: is that node Lexington? Wait, the label says "Lexington" is the node connected with 9 miles from Stafford? Wait, the graph shows: Springfield ---14.4---> Stafford ---9---> Lexington? Wait, no, the original image: "Springfield" connected to "Stafford" (14.4), "Springfield" connected to "Belmont" (7.6), "Belmont" connected to "Stafford" (7.5), "Stafford" connected to a node (maybe Lexington? No, the label says "Lexington" is the node after the 9 - mile path? Wait, the text says "how long is the shortest route from Springfield to Lexington". Let's re - calculate the two possible routes from Springfield to Stafford first:

Springfield to Stafford:
- Direct: \(14.4\) miles.
- Through Belmont: \(7.6+7.5 = 15.1\) miles. So the shorter one is \(14.4\) miles.

Then from Stafford to Lexington: the distance is \(9\) miles (since that's the path to Lexington, the other path from Stafford goes to Brookfield, which is not part of the route to Lexington). So total distance: \(14.4 + 9=23.4\)? Wait, but wait, maybe I made a mistake. Wait, let's check the numbers again. Wait, \(7.6+7.5 = 15.1\), which is more than \(14.4\), so Springfield to Stafford is \(14.4\). Then Stafford to Lexington is \(9\). So \(14.4 + 9 = 23.4\). Wait, but maybe the path from Stafford to Lexington is not 9? Wait, the graph: "Stafford" ---9 mi---> a node, and that node is "Lexington"? Or is "Lexington" connected to that node? Wait, the original problem's graph: "Springfield", "Belmont", "Stafford", then "Stafford" ---9 mi---> "Lexington"? No, the label says "Lexington" is the node after the 9 - mile path? Wait, the user's image: "Springfield" (red dot), "Belmont" (red dot), "Stafford" (red dot), then "Stafford" ---9 mi---> a red dot labeled "Lexington"? Wait, no, the text says "Brookfield" is another node. Wait, maybe the correct routes are:

Route 1: Springfield - Stafford (14.4) - (9) - Lexington. Total: 14.4 + 9 = 23.4

Route 2: Springfield - Belmont (7.6) - Stafford (7.5) - (9) - Lexington. Total: 7.6+7.5 + 9 = 24.1

Route 3: Is there a path from Springfield to Lexington without going through Stafford? No, from the graph, Springfield is connected to Stafford and Belmont, Belmont to Stafford, Stafford to Lexington (and to Brookfield). So the two main routes are the two above. So the shortest is 23.4? Wait, but wait, maybe I misread the distance from Stafford to Lexington. Wait, the graph: "Stafford" ---9 mi---> a node, and that node is "Lexington"? Or is "Lexington" connected to that node with another distance? Wait, the original image: "Brookfield" is connected to the node after Stafford with 4.5 mi, but "Lexington" is the node before Brookfield? Wait, no, the labels: "Springfield", "Belmont", "Stafford", then "Stafford" ---9 mi---> "Lexington"? No, the user's problem says "how long is the shortest route from Springfield to Lexington". Let's re - check the arithmetic:

First, compare the two paths from Springfield to Stafford:

Path 1: Springfield to Stafford directly: \(14.4\) miles.

Path 2: Springfield to Belmont (\(7.6\)) + Belmont to Stafford (\(7.5\)) = \(7.6 + 7.5=15.1\) miles.

Since \(14.4<15.1\), we choose the direct path from Springfield to Stafford (\(14.4\) miles).

Then, from Stafford to Lexington: the distance is \(9\) miles (as per the graph, the path from Stafford to Lexington is \(9\) miles, the other path from Stafford goes to Brookfield, which is not part of the route to Lexington).

So total distance: \(14.4 + 9 = 23.4\) miles. Wait, but wait, maybe the path from Stafford to Lexington is not 9? Wait, maybe I made a mistake. Wait, let's calculate again. Wait, \(7.6+7.5 = 15.1\), \(14.4\) is less than \(15.1\), so Springfield to Stafford is \(14.4\). Then Stafford to Lexington is \(9\), so \(14.4 + 9=23.4\). Yes, that seems correct.

Answer:

\(23.4\)