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Coconut palm trees can reach heights of up to 100 feet. Suppose you are lying on the beach at a distance of 60 feet from a 48-ft tall palm tree. What is the angle of elevation from your position to the top of the tree? Round to the nearest tenth if necessary.

A. [tex]$128.7^{\circ}$[/tex]
B. [tex]$53.1^{\circ}$[/tex]
C. [tex]$36.9^{\circ}$[/tex]
D. [tex]$38.7^{\circ}$[/tex]


Sagot :

Let's determine the angle of elevation from your position to the top of the 48-foot tall palm tree.

First, let's define the problem parameters:
- Distance from the tree (adjacent side of the right triangle): 60 feet
- Height of the tree (opposite side of the right triangle): 48 feet

We can use trigonometry to find the angle of elevation. Specifically, the tangent function, which relates the opposite side to the adjacent side in a right triangle, is useful here:

[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]

Substitute the known values into the equation:

[tex]\[ \tan(\theta) = \frac{48}{60} \][/tex]

Now calculate the ratio:

[tex]\[ \tan(\theta) = 0.8 \][/tex]

To find the angle [tex]\(\theta\)[/tex], we take the inverse tangent (arctan) of 0.8:

[tex]\[ \theta = \arctan(0.8) \][/tex]

Using a calculator or trigonometric table to find the inverse tangent, we get:

[tex]\[ \theta \approx 38.7^\circ \][/tex]

So, the angle of elevation from your position to the top of the tree, rounded to the nearest tenth, is [tex]\(38.7^\circ\)[/tex].

Therefore, the correct answer is:
d. [tex]\( 38.7^\circ \)[/tex]
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