Chapter 6.4, Problem 92AYU

To determine

To solve: equation for the sine curve that fits the opening and to find the height of the tunnel at edge of the road.

Answer to Problem 92AYU

  $y=15\sin \left(\frac{2\pi }{28}x\right)$

  $y=10.606ft$

Explanation of Solution

Amplitude of function = highest point of cave

  $A=15$

Period of curve = opening $=28$

  $\begin{array}{l}\frac{2\pi }{k}=28\\ k=\frac{2\pi }{28}\\ y=A\sin kx\\ y=15\sin \left(\frac{2\pi }{28}x\right)\end{array}$

Here at the edge of the road,

  $\begin{array}{l}x=7\\ y=15\sin \left(\frac{\pi }{28}\times 7\right)\\ y=15\times 0.7071\\ y=10.606ft\end{array}$