Question

1. points p and q are on the north and south rims, respectively, of the grand canyon, with q directly south of p. point r is located 780 meters west of p. the measure of ∠prq is 84°40. find pq, the width of the canyon.

Explanation:

Step1: Convert angle to decimal

First, convert $84^{\circ}40'$ to decimal degrees. Since $1^{\circ}=60'$, then $40'=\frac{40}{60}\approx0.6667^{\circ}$, so $84^{\circ}40'\approx84.6667^{\circ}$.

Step2: Use tangent function

In right - triangle $PRQ$, we know that $\tan\angle{PRQ}=\frac{PQ}{PR}$. Given $PR = 780$ meters and $\angle{PRQ}\approx84.6667^{\circ}$.
We have $PQ=PR\times\tan\angle{PRQ}$.
Substitute the values: $PQ = 780\times\tan(84.6667^{\circ})$.
Since $\tan(84.6667^{\circ})\approx10.585$, then $PQ=780\times10.585 = 8256.3$ meters.

Answer:

$8256.3$ meters