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An ant looks up to the top of a tree at an angle of elevation of 40°. If the ant is 30 feet from the base of the tree, what is the height of the tree, to the nearest tenth?

A. 19.3 feet
B. 23.0 feet
C. 25.2 feet
D. 35.8 feet

Please select the best answer from the choices provided.


Sagot :

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.