Question: When the angle of elevation of the sun is $62^\circ$, a telephone pole that is tilted at an angle of $8^\circ$ directly away from the sun casts a shadow $21$ feet long. Determine the length of the pole to the nearest foot.
1. **State the problem:** We need to find the length of a telephone pole that is tilted $8^\circ$ away from the sun when the sun's elevation angle is $62^\circ$ and the shadow length is $21$ feet.
2. **Understand the setup:** The sun's rays form a $62^\circ$ angle with the horizontal ground. The pole is tilted $8^\circ$ away from the sun's direction, so the angle between the pole and the ground is $62^\circ - 8^\circ = 54^\circ$.
3. **Identify the right triangle:** The shadow forms the adjacent side to the pole's angle with the ground, and the pole length is the hypotenuse.
4. **Use the cosine function:** For angle $\theta = 54^\circ$, the cosine relates the adjacent side (shadow length) to the hypotenuse (pole length $b$):
$$\cos(54^\circ) = \frac{\text{shadow length}}{\text{pole length}} = \frac{21}{b}$$
5. **Solve for $b$:**
$$b = \frac{21}{\cos(54^\circ)}$$
6. **Calculate $\cos(54^\circ)$:**
$$\cos(54^\circ) \approx 0.5878$$
7. **Calculate $b$:**
$$b = \frac{21}{0.5878} \approx 35.74$$
8. **Round to the nearest foot:**
$$b \approx 36$$
**Final answer:** The length of the pole is approximately **36** feet.