Question

the angle of elevation to a nearby tree from a point on the ground is measured to be 32°. how tall is the tree if the point on the ground is 71 feet from the bottom of the tree? round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Set up the trigonometric relationship

We have a right - triangle where the adjacent side to the angle of elevation (\(\theta = 32^{\circ}\)) is \(x = 71\) feet and the opposite side is the height of the tree \(h\). Using the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan(32^{\circ})=\frac{h}{71}\).

Step2: Solve for \(h\)

Multiply both sides of the equation \(\tan(32^{\circ})=\frac{h}{71}\) by \(71\). We get \(h = 71\times\tan(32^{\circ})\).
Since \(\tan(32^{\circ})\approx0.6249\), then \(h=71\times0.6249 = 44.3679\).

Answer:

\(44.4\) feet