To determine the angle of elevation from the tip of the shadow to the top of a 14-foot tree casting an 18-foot shadow, we'll use trigonometry in the following steps:
1. Identify the Right Triangle:
- The tree and its shadow form a right-angled triangle.
- The height of the tree (14 feet) is the vertical leg.
- The length of the shadow (18 feet) is the horizontal leg.
2. Use the Tangent Function:
- In a right triangle, the tangent of an angle (θ) is the ratio of the opposite side to the adjacent side.
- [tex]\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)[/tex]
- Here, the opposite side is the height of the tree, and the adjacent side is the length of the shadow.
3. Calculate [tex]\(\tan(\theta)\)[/tex]:
- [tex]\(\tan(\theta) = \frac{14}{18}\)[/tex]
4. Find the Angle:
- To find the angle of elevation, we need to take the arctan (inverse tangent) of the ratio.
- [tex]\(\theta = \arctan\left(\frac{14}{18}\right)\)[/tex]
5. Convert to Degrees:
- The result of [tex]\(\arctan\)[/tex] will be in radians. We need to convert this value to degrees.
- [tex]\( \theta \text{ (in degrees)} = \theta \text{ (in radians)} \times \left(\frac{180}{\pi}\right) \)[/tex]
6. Calculate the Angle in Degrees:
- After calculating [tex]\(\arctan\left(\frac{14}{18}\right)\)[/tex], we convert the result to degrees.
- This gives approximately 37.8749836510982 degrees.
7. Round the Result:
- Finally, round the angle to the nearest tenth.
- 37.8749836510982 degrees rounds to 37.9 degrees.
Given the choices:
a. 37.9°
b. 38.9°
c. 51.1°
d. 52.1°
The best answer is:
a. 37.9°
