Question
A tree casts a shadow 21 m long. The angle of elevation of the sun is \( 51^{\circ} \). What is the helght of the tree?
Ask by Barker Ray. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The height of the tree is approximately 25.93 meters.
Solution
1. We start with the relation for the tangent of the angle of elevation:
\[
\tan(51^\circ) = \frac{\text{height}}{\text{shadow length}}
\]
2. Substitute the shadow length:
\[
\tan(51^\circ) = \frac{h}{21}
\]
3. Solve for \( h \):
\[
h = 21 \tan(51^\circ)
\]
4. Using a calculator, compute \( \tan(51^\circ) \approx 1.2349 \):
\[
h \approx 21 \times 1.2349 \approx 25.93 \text{ m}
\]
Thus, the height of the tree is approximately \( 25.93 \) m.
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Simplify this solution Bonus Knowledge
To find the height of the tree, we can use the tangent function in trigonometry, which relates angles to opposite and adjacent sides in a right triangle. If the height of the tree is represented as \( h \), the length of the shadow is the adjacent side (21 m), and the angle of elevation of the sun is \( 51^{\circ} \): \[ \tan(51^{\circ}) = \frac{h}{21} \] Rearranging gives: \[ h = 21 \times \tan(51^{\circ}) \] Calculating this, we find \( h \approx 21 \times 1.2349 \approx 25.94 \) m. So, the height of the tree is approximately 25.94 meters. That’s a tall tree basking in the sun!
