Question

9 formula 0.5 points
an observer stands on the bank of a river, and looks directly across the river to a tree on the opposite bank. the angle of elevation from the feet of the observer to the top of the tree is 48 degrees.
if the tree is 23 feet tall, how wide is the river?
be sure your calculator is in deg mode, and use the proper trig function on your calculator in the computation.
round your answer to the nearest tenth of a foot, but do not include \ft\ with your response.
answer
20.7

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the height of the tree (opposite side to the angle of elevation) is 23 feet, the width of the river is the adjacent side (\(x\)) to the angle of elevation (48 degrees), and we can use the tangent function. The tangent of an angle in a right triangle is defined as \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). So, \(\tan(48^\circ)=\frac{23}{x}\).

Step2: Solve for \(x\)

We can rearrange the formula to solve for \(x\): \(x = \frac{23}{\tan(48^\circ)}\). Now, we calculate \(\tan(48^\circ)\) (make sure the calculator is in degree mode). \(\tan(48^\circ)\approx1.1106\). Then, \(x=\frac{23}{1.1106}\approx20.7\).

Answer:

20.7