Question
A flagpole casts a shadow that is 137 feet long. The angle of elevation from the end of the shadow to the top of the flagpole is 25 degrees. How tall is the flagpole?
Answers
GPT-4o mini
To find the height of the flagpole, we can use trigonometry. Specifically, we can use the tangent function, which relates the angle of elevation to the opposite side (height of the flagpole) and the adjacent side (length of the shadow).
Let \( h \) be the height of the flagpole. Using the tangent of the angle of elevation:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case, the angle \( \theta \) is 25 degrees, the opposite side is the height \( h \) of the flagpole, and the adjacent side is the length of the shadow, which is 137 feet. Therefore, we can write:
\[
\tan(25^\circ) = \frac{h}{137}
\]
Now, to find the height \( h \), we can rearrange this equation:
\[
h = 137 \cdot \tan(25^\circ)
\]
Next, we need to calculate \( \tan(25^\circ) \). Using a calculator:
\[
\tan(25^\circ) \approx 0.4663
\]
Now, substituting this value back into the equation for \( h \):
\[
h = 137 \cdot 0.4663 \approx 64.0
\]
Thus, the height of the flagpole is approximately:
\[
\boxed{64} \text{ feet}
\]
Let \( h \) be the height of the flagpole. Using the tangent of the angle of elevation:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case, the angle \( \theta \) is 25 degrees, the opposite side is the height \( h \) of the flagpole, and the adjacent side is the length of the shadow, which is 137 feet. Therefore, we can write:
\[
\tan(25^\circ) = \frac{h}{137}
\]
Now, to find the height \( h \), we can rearrange this equation:
\[
h = 137 \cdot \tan(25^\circ)
\]
Next, we need to calculate \( \tan(25^\circ) \). Using a calculator:
\[
\tan(25^\circ) \approx 0.4663
\]
Now, substituting this value back into the equation for \( h \):
\[
h = 137 \cdot 0.4663 \approx 64.0
\]
Thus, the height of the flagpole is approximately:
\[
\boxed{64} \text{ feet}
\]