Sure, let's go through the steps to solve this problem.
1. Understand the Situation:
- An ant is located 30 feet away from the base of a tree.
- The angle of elevation from the ant's position to the top of the tree is 40°.
2. Identify the Relevant Trigonometric Function:
- We need to find the height of the tree (opposite side in the right triangle).
- Given the angle of elevation and the distance from the base (adjacent side), we can use the tangent function:
[tex]\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Therefore:
[tex]\[
\text{opposite} = \text{adjacent} \times \tan(\text{angle})
\][/tex]
3. Apply the Values:
- The adjacent side (distance from the base) is 30 feet.
- The angle of elevation is 40°.
4. Calculate the Height of the Tree:
- First, convert 40° to radians since trigonometric functions in most calculators (and programming languages) require the angle in radians.
- Compute:
[tex]\[
\text{height} = 30 \times \tan(40^\circ)
\][/tex]
- Using the tangent of 40°:
[tex]\[
\text{height} \approx 30 \times \tan(40^\circ)
\][/tex]
- This calculation gives approximately 25.2 feet when rounded to the nearest tenth.
5. Final Answer:
- Hence, the height of the tree is approximately 25.2 feet.
Therefore, the best answer from the choices provided is:
c. 25.2 feet.
