Question

mariela is standing in a building and looking out of a window at a tree. the tree is 20 feet away from mariela. marielas line of sight to the top of the tree creates a 42° angle of elevation, and her line of sight to the base of the tree creates a 31° angle of depression. what is the height, in feet, of the tree? enter your answer in the box.

Explanation:

Step1: Calculate the height above the horizontal line

Use the tangent function for the \(31^{\circ}\) angle. The formula for tangent is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 31^{\circ}\) and the adjacent side is \(20\) ft. So, the height above the horizontal line \(h_1=20\times\tan(31^{\circ})\).
Since \(\tan(31^{\circ})\approx0.60086\), then \(h_1 = 20\times0.60086=12.0172\) ft.

Step2: Calculate the height below the horizontal line

Use the tangent function for the \(42^{\circ}\) angle. The formula for tangent is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 42^{\circ}\) and the adjacent side is \(20\) ft. So, the height below the horizontal line \(h_2=20\times\tan(42^{\circ})\).
Since \(\tan(42^{\circ})\approx0.9004\), then \(h_2 = 20\times0.9004 = 18.008\) ft.

Step3: Calculate the total height of the tree

The total height of the tree \(H=h_1 + h_2\).
\(H=12.0172+18.008=30.0252\approx30\) ft.

Answer:

\(30\)