A 12-foot tree casts a 22-foot shadow. Find the angle of elevation from the tip of the shadow to the top of the tree. Round the answer to the nearest tenth.

A. [tex]$28.6^{\circ}$[/tex]
B. [tex]$33.1^{\circ}$[/tex]
C. [tex]$56.9^{\circ}$[/tex]
D. [tex]$61.4^{\circ}$[/tex]

Please select the best answer from the choices provided:
A
B
C
D



Answer :

To find the angle of elevation from the tip of the shadow to the top of the tree, given a 12-foot tree that casts a 22-foot shadow, follow these steps:

1. Identify the given values:
- The height of the tree (opposite side): [tex]\( 12 \)[/tex] feet.
- The length of the shadow (adjacent side): [tex]\( 22 \)[/tex] feet.

2. Use the trigonometric function tangent, which relates the opposite side and the adjacent side of a right triangle:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Here, [tex]\(\theta\)[/tex] is the angle of elevation we need to find.

3. Set up the equation using the known values:
[tex]\[ \tan(\theta) = \frac{12}{22} \][/tex]

4. Calculate the arctangent (inverse tangent) of [tex]\( \frac{12}{22} \)[/tex] to find [tex]\( \theta\)[/tex]:
[tex]\[ \theta = \arctan\left(\frac{12}{22}\right) \][/tex]

5. Convert this angle in radians to degrees:
- Use a calculator to find the angle in radians:
[tex]\[ \theta \approx 0.499 \text{ radians} \][/tex]
- Convert radians to degrees:
[tex]\[ \theta \approx 28.610^{\circ} \][/tex]

6. Round the angle to the nearest tenth:
[tex]\[ 28.610^{\circ} \approx 28.6^{\circ} \][/tex]

Therefore, the angle of elevation from the tip of the shadow to the top of the tree is approximately [tex]\( 28.6^{\circ} \)[/tex].

Thus, the best answer from the choices provided is:

a. [tex]\( 28.6^{\circ} \)[/tex]