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]
