Question

the angle of elevation to a nearby tree from a point on the ground is measured to be 42°. how tall is the tree if the point on the ground is 43 feet from the bottom of the tree? round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Use the tangent function

In a right - triangle (formed by the tree, the ground, and the line of sight), the tangent of an angle of elevation \(\theta\) is given by \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 42^{\circ}\), the adjacent side \(a = 43\) feet, and the opposite side (height of the tree) is \(x\). So, \(\tan(42^{\circ})=\frac{x}{43}\).

Step2: Solve for \(x\)

We can rewrite the equation as \(x = 43\times\tan(42^{\circ})\). Using a calculator, \(\tan(42^{\circ})\approx0.9004\). Then \(x=43\times0.9004 = 38.7172\).

Answer:

\(38.7\) feet