Scorpion Gulch and Western Railway are preparing to build a new line through Rolling Mountains. They have hired you to do some calculations for tunnels and bridges needed on the line. You set up a Cartesian coordinate system with its origin at the entrance to the tunnel through Bald Mountain. Your surveying crew finds that the mountain rises 250 m above the level or the track and that the next valley goes down 50 m bellow the level of the track. The cross section of the mountain and the valley is roughly sinusoidal with a horizontal distance of 700 m from the top of the mountain to the bottom of the valley.

Question

Scorpion Gulch and Western Railway are preparing to build a new line through Rolling Mountains. They have hired you to do some calculations for tunnels and bridges needed on the line. You set up a Cartesian coordinate system with its origin at the entrance to the tunnel through Bald Mountain. Your surveying crew finds that the mountain rises 250 m above the level or the track and that the next valley goes down 50 m bellow the level of the track. The cross section of the mountain and the valley is roughly sinusoidal with a horizontal distance of 700 m from the top of the mountain to the bottom of the valley. 
a. Write an equation expressing the vertical distance y from the track to the surface of the mountain or valley in terms of the horizontal distance x from the tunnel entrance. 
b. How long will the tunnel be?
c. How long will the bridge be?
d. The company thinks it might be cheaper to build the line if the entire project is raised to y=20, thus making the tunnel shorter and the bridge longer. Find the new values of x at the ends of the tunnel and bridge. Then find the new lengths of each.

Rob Gronkowski · Answer

Honestly, this problem is such a joke that I don't even think Mahatma Ghandi could solve it.