Question

a tree on one side of a river is due west of a rock on the other side of the river. from a stake 19.0 yards north of the rock, the bearing of the tree is s 18.5° w. how far is it from the rock to the tree? round to at least the nearest tenth of a yard.

Explanation:

Step1: Identify the right - triangle

We have a right - triangle where the vertical side (opposite side with respect to the given angle) is the distance from the rock to the stake, which is $y = 19.0$ yards, and the angle of depression from the stake to the tree is $\theta=18.5^{\circ}$. We want to find the horizontal side (adjacent side with respect to the given angle) $x$, which is the distance from the rock to the tree.

Step2: Use the tangent function

The tangent of an angle in a right - triangle is defined as $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\tan(18.5^{\circ})=\frac{19.0}{x}$.

Step3: Solve for $x$

We can re - arrange the equation $\tan(18.5^{\circ})=\frac{19.0}{x}$ to get $x=\frac{19.0}{\tan(18.5^{\circ})}$.
We know that $\tan(18.5^{\circ})\approx0.3365$. Then $x=\frac{19.0}{0.3365}\approx56.5$ yards.

Answer:

$56.5$ yards