Question
Clyde is looking at a flagpole. He looks up at the flag at a 50° angle of elevation, and the diagonal distance from his face to the top of the pole is 25 feet. Find the height of the flagpole given that Clyde’s eye level is 6 feet above the ground. Round the answer to the nearest tenth. A full point answer will include: Equation that shows the correct selection of sine, cosine or tangent. Equation also shows the correct ratio/fraction order. Solution is correct. Solution is accurately labeled in units of measurement and rounded to the nearest tenth. (4 points)
Answers
GPT-4o mini
To find the height of the flagpole, we can use the information provided and trigonometry, specifically the sine function.
1. **Define the right triangle involved:**
- Let \( h \) be the height of the flagpole above Clyde's eye level.
- The total height of the flagpole from the ground is then \( h + 6 \) feet.
- The angle of elevation from Clyde's eye level to the top of the flagpole is \( 50^\circ \).
- The diagonal distance (hypotenuse) from Clyde's eye to the top of the flagpole is \( 25 \) feet.
2. **Using sine:**
The sine function relates the angle, the opposite side (height above eye level), and the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
For our problem:
\[
\sin(50^\circ) = \frac{h}{25}
\]
3. **Setting up the equation:**
\[
h = 25 \sin(50^\circ)
\]
4. **Calculating \( h \):**
To find \( h \):
\[
h = 25 \sin(50^\circ)
\]
Using a calculator, we find \( \sin(50^\circ) \approx 0.7660 \):
\[
h \approx 25 \times 0.7660 \approx 19.15 \text{ feet}
\]
5. **Total height of the flagpole:**
The total height of the flagpole from the ground is:
\[
H = h + 6 \text{ feet} = 19.15 \text{ feet} + 6 \text{ feet} = 25.15 \text{ feet}
\]
6. **Final Answer:**
Rounding to the nearest tenth:
\[
\text{Height of the flagpole} \approx 25.2 \text{ feet}
\]
**Summary:**
The height of the flagpole is approximately **25.2 feet**.
1. **Define the right triangle involved:**
- Let \( h \) be the height of the flagpole above Clyde's eye level.
- The total height of the flagpole from the ground is then \( h + 6 \) feet.
- The angle of elevation from Clyde's eye level to the top of the flagpole is \( 50^\circ \).
- The diagonal distance (hypotenuse) from Clyde's eye to the top of the flagpole is \( 25 \) feet.
2. **Using sine:**
The sine function relates the angle, the opposite side (height above eye level), and the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
For our problem:
\[
\sin(50^\circ) = \frac{h}{25}
\]
3. **Setting up the equation:**
\[
h = 25 \sin(50^\circ)
\]
4. **Calculating \( h \):**
To find \( h \):
\[
h = 25 \sin(50^\circ)
\]
Using a calculator, we find \( \sin(50^\circ) \approx 0.7660 \):
\[
h \approx 25 \times 0.7660 \approx 19.15 \text{ feet}
\]
5. **Total height of the flagpole:**
The total height of the flagpole from the ground is:
\[
H = h + 6 \text{ feet} = 19.15 \text{ feet} + 6 \text{ feet} = 25.15 \text{ feet}
\]
6. **Final Answer:**
Rounding to the nearest tenth:
\[
\text{Height of the flagpole} \approx 25.2 \text{ feet}
\]
**Summary:**
The height of the flagpole is approximately **25.2 feet**.