So far in our quest to find the most economical location for the new stadium, we can draw the following conclusions:
- There are two main route options from each city to each center: a direct route with all newly paved roads or an indirect route that maximizes use of existing roads while minimizing the distance of newly paved roads.
- The shortest total distance between two points will be achieved by a straight road between them.
- The shortest distance from a triangle center to an existing road will be a road perpendicular to the existing road.
Step 2: Model with GeoGebra
Wit these ideas in mind, we can use GeoGebra to construct the potential pathways from each city to each center. For the sake of clarity, one GeoGebra sketch has been used showing the pathways from each city (vertex) to each center (stadium). One unit represents one mile. In all of the sketches, the roads to the circumcenter, centroid, incenter, and orthocenter are blue, pink, green, and orange, respectively. Existing roads (perimeter of triangle) are not highlighted a specific color, but one can logically deduce whether an existing road was used in a given route.
Routes from Durham:
Routes from Raleigh:
Routes from Chapel Hill:
Step 3: Calculate and Organize Costs
Now that we have a visual of possible roads, we can calculate the cost of paving each route by adding together the cost of paving any necessary new roads ($125,000/mi.) plus the cost of repaving any existing roads ($50,000/mi.) for each route. We can organize the calculations in a spreadsheet, like the one created with Numbers below. The spreadsheet is organized such that the types of routes (direct/indirect) are in rows, grouped and color-coded by city from which the route starts. The columns represent the triangle centers, which are the destinations for any given route. In each column, the cheapest route from a given city is highlighted with the color assigned to that city. The last row of the chart shows the cheapest total cost of constructing roads to the corresponding stadium location. Each cell in this last row was calculated by adding together the cheapest route options from each city (highlighted cells) then subtracting the cost to (re)pave the roads they have in common. By subtracting the cost to (re)pave common roads, we avoid duplicating any expenses in our total.
Step 4: Analyze the Data
Comparing total costs in each of the columns in the last row, we see that building the stadium at the circumcenter will result in the cheapest (and therefore most economical) combination of road construction. This cell is highlighted blue in the spreadsheet above, showing that the total cost of (re)paving roads to a stadium located at the circumcenter would be $2,491,250.
Step 5: Construct the Solution
The construction made with GeoGebra shown below illustrates the roads necessary for building the new stadium at the circumcenter. Repaved roads are highlighted blue, while the new paved road is highlighted pink.
This construction depicting our conclusion exemplifies how technology can be used to consider the reasonableness of our results. Since we know that the cost of repaving roads ($50,000/mi.) is significantly cheaper than the cost of paving new roads ($125,000/mi.), it makes sense that our solution would make the most use out of existing highways while minimizing the distance of new roads needed. Furthermore, we know that the more roads each route has in common with the other routes, the less roads have to be (re)paved and the more money we save. Our solution reflects this in that all three routes share one newly paved road (which happens to be the shortest possible newly paved road from a center to a highway), and the routes from Durham and Chapel Hill share the stretch of existing highway that extends from Chapel Hill to the intersection with the new road.
With the problem solved, only two questions remain: what implications does this example have for a classroom, and which baseball team will play at the new stadium?