Question

Finding the Length of a Ski Lift Consult the figure. To find the length of the span of a proposed ski lift from P to Q, a surveyor measures $\angle$DPQ to be 25° and then walks off a

distance of 1000 feet to R and measures $\angle$PRQ to be 15°. What is the distance from P to Q?

Short answer

The length of Ski lift is 1490.48 feet.

Step-by-step solution

Step 1. Given Information

A ski lift is to be built from the bottom of the mountain P to the top Q.

A surveyor measures the angle of elevation from his position P to the top of the mountain Q is $\angle DPQ={25}^{\circ }.$

He then walks 1000 feet away from the mountain to Point R. He measure the angle of elevation from this position R to the top of the mountain Q is $\angle PRQ={15}^o$

The following scales models the problem

Step 2. Find the measure of angle ∠QPR,∠QPD and ∠QPR complete a straight line,

$$\begin{gathered} \angle QPD+\angle QPR=180 \\ 25+\angle QPR=180 \\ \angle QPR={155}^{\circ } \end{gathered}$$

and$$\begin{gathered} \angle PQR+\angle QPR+\angle PRQ=180 \\ \angle PQR+155+15=180 \\ \angle PQR={10}^{\circ } \end{gathered}$$

Step 3. Finding the length of Ski lift

$$\begin{gathered} \frac{\sin \left(\angle PRQ\right)}{PQ}=\frac{\sin \left(\angle PQR\right)}{PR} \\ \frac{\sin {15}^{\circ }}{PQ}=\frac{\sin {10}^{\circ }}{1000} \\ PQ=\sin {15}^{\circ }\times \frac{1000}{\sin \left({10}^{\circ }\right)} \\ PQ=1490.48feet \end{gathered}$$