Sue Pinion, Sandy McAdams, Lisa Stueve

## A bridge is built one mile long and is built with a material which expands to accomodate for the heat factor. When the bridge is fully expanded, the length of the bridge is 5282 feet. How high does the bridge rise?

The problem could be considered to be the arc of a circle as shown below. The equations can be generated from this diagram using the formulas for the length of the arc of a circle and trigonometric functions.

The solution to this system is not obvious. We created a spreadsheet to substitute a value for n, evaluate the value of sin (n), and compute the value of r in each equation. Then the spreadsheet calculated the difference in the values of r for each equation. When we found a value for n which created a value for r in each equation with differences close to zero, we continued to add significant digits to our n value in order to get closer to a zero difference.

After deciding that the value of n =.04766, we calculated the radius from the above equations and found r =55406.369. Then we had to calculate x in the picture below:

This value was calculated using the value of r and the pythagorean theorem and found to equal 55343.43796. After subtracting r, the bridge is found to rise vertically 62.93 feet.

Extension:
Suppose a span of 43 feet must have a bridge built across it. The bridge has a vertical rise of 9 inches. How long is the bridge?

We used the pythagorean theorem and solved for x=307.79166 feet. This means r= 308.5166 feet and n=.069739163. Therefore s=21.51568 and the entire bridge length is approximately 43.0313 feet.

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