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Monday, February 15, 2010

This problem is to figure out the lengths of 3 different routes bisecting the circular Caldera rim and valley. The pertinent information is that the Kid’s village is in the exact center of the circular valley which has a 100km radius. The northern-most point on the rim is Sentry Point 1. Hope Lake is 30km south of there. Sentry Point 2 is due west of Hope Lake. The Adult’s village is 15km south of Sentry Point 2. We need to figure out the direct overland distance (“as the crow flies”) from Sentry Point 1 to the Adult’s village. We also need to figure out the distance on the usual adult route from Sentry Point 1 to the Adult village, going south to Hope Lake, then West to Sentry Point 2 and south to the Adult village. Finally, we also need to calculate the distance if you went on the rim from Sentry Point 1 to Sentry Point 2 and then South to the village. The main math topic imbedded in this problem is geometry.

My first step in solving this problem was to draw out the situation on a diagram/map. Heres what it looks like:

I decided to figure out the easiest thing first which was the usual adult route, since I knew the distance from Sentry Point1 to Hope Lake is 30km and the distance from Sentry Point 2 to Adult’s village is 15km. I needed to figure out the distance from the Hope Lake to Sentry Point 2. I realized that this length represented one side of a right triangle formed by Hope Lake, Kids Village and Sentry Point 2. I knew the distance from Hope Lake to Kids village was 70km. I also knew the distance from Kid’s village to Sentry Point 2 was 100km, because it is the radius of the circle. I used the Pythagorean theorem as follows:
4900 + b2 = 1000
b2 = 5100

So I added the 3 lengths and got the total usual adult path was 116km.

Next, I decided to figure out “as the crow flies” distance. I realized that this distance was the hypotenuse of yet another triangle formed by Sentry Point 1, Adult’s village and a point 15km due south of Hope Lake. I knew the lengths of two of the sides were 45km and 71km. I used Pythagorean theorem (see above) to figure out that the hypotenuse was 84km.

Finally, I needed to figure out the distance in the around-the-rim route. I found a formula online that was based on angles that was like this:


I needed to figure out the angle of the two lines between Kids village and Sentry Point 1 and Kids village and Sentry Point 2. I decided to use a protractor to measure the angle after measuring my lines carefully. Here’s what my diagram looked like:

With the protractor I measured the angle at 45 degrees. Plugging the numbers into the formula. I got:

200km (the diameter) * 3.14159 * 45/360 = 78.5km or 79

I then added the extra 15km south to the Adult’s Village from Sentry Point 2 to 79 and got a total of 94km. The resources I used to solve the problem were my Mac, keynote (for diagrams), the Internet (for formulas) and my Mom.

My solution is that the longest route is the usual adult route at 116km, the next longest is the rim route at 94km, and the shortest was the “as the crow flies” at 84km. I think my answers are correct. I do not think there are other answers. It makes sense that the shortest route would be the most direct (a line between 2 points).

This problem was hard because it took a while to discover the triangles that could lead to the answer. I also wasn’t sure using the protractor was the only way to get the arc length, but it was the only way I could figure out. Because I used a protractor, my answer might be less precise than if I had just used a formula. I think I managed my time poorly, but it was all I could do as I had a lot to do. 

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