Legend+of+Zelda

Workers are designing a road to go over a mountain. They want the road to have a maximum grade of 12%. The lowest gap in the mountain that they are going to go over has an elevation of 1623 feet above sea level. The road will start at an elevation of 412 feet above sea level and meet up with a road on the other side at an elevation of 631 feet above sea level. What is the minimum length of the road to meet these requirements? What is the average angle of elevation for this road?
 * Emily Allport, Nichole Barker, Reggie Watts, Cole Martin


 * First, the problem says that the maximum grade of the road is 12%. To find the angle created by that grade, take 180° and multiply it by 12% (or .12). The angle is 21.6°.
 * Second, since the problem says that the lowest gap is 1623 ft, we added one to make the tallest peaks 1624 ft tall.
 * Next, find the length of the road for part a. Since the mountain is 1624 ft tall and the road starts at 412 ft, subtract those two numbers to get 1212 ft. This is the side opposite of the angle. Then do 1212 divided by sin(21.6°) to find the length of a. It is 3292.36 ft.
 * After that, find the lengths of b and c. These triangles are identical because of their dimensions. First, the vertical component is known to be 1 ft (1624-1623). The angles at the bottom corners of the triangles are both 21.6°. Use 1 divided by sin(21.6°) to find the b and c components of 2.72 ft. (added together to be 5.44 ft).
 * Then, to find the length of d, take 1624 (the height of the mountain) minus 631 (the height of the end of the road). This equals 993 ft. Then use this information to find d by dividing 993 by sin(21.6°). This equals 2697.46 ft.
 * Lastly add a + b + c + d to get a final answer of 5995.26 ft.
 * Next we needed to find the average angle of elevation. To do this, first find the bottom sides of all four triangles by using cos(21.6°) and the opposites sides found earlier. Answers: 3058.64 + 2.526 + 2.526 + 2508.03. This equals 5571.72 ft.
 * Then, since the end of the road is at 631ft and the beginning is at 412 ft, subtract the two to get the difference and find the opposite side. It is 219 ft.
 * Lastly, use inverse tangent of (219 divided by 5771.72) to get a final answer of 2.2°.



Going by the maximum 12% gradient directly to the mountain, and then 12% down to the end of the road would take 18,490ft of road. The average angle of elevation is 6.84 degrees.