Question
Deshaun is looking up at a flag that is 50 feet away from him at an angle of elevation from ground level of 35° . What is the flagpole’s height, x, and the distance between Deshaun and the top of the flagpole, y?
Answers
GPT-4o mini
To solve for the height of the flagpole, \( x \), and the distance \( y \) between Deshaun and the top of the flagpole, we can use trigonometric relationships.
1. **Finding the height of the flagpole (\( x \))**:
We can use the tangent function, which relates the angle of elevation to the opposite side (height of the flagpole) and the adjacent side (distance to the base of the flagpole).
The formula is:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Where:
- \( \theta = 35^\circ \)
- Opposite = height of the flagpole \( x \)
- Adjacent = distance to the base of the flagpole = 50 feet
So we rewrite the equation as:
\[
\tan(35^\circ) = \frac{x}{50}
\]
Now solve for \( x \):
\[
x = 50 \tan(35^\circ)
\]
Using a calculator, find \( \tan(35^\circ) \):
\[
\tan(35^\circ) \approx 0.7002
\]
Thus,
\[
x \approx 50 \times 0.7002 \approx 35.01 \text{ feet}
\]
So the height of the flagpole \( x \) is approximately **35.01 feet**.
2. **Finding the distance from Deshaun to the top of the flagpole (\( y \))**:
To find \( y \), we can use the sine function, which relates the angle of elevation to the opposite side (height of the flagpole) and the hypotenuse (distance from Deshaun to the top of the flagpole):
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
So we rewrite the equation as:
\[
\sin(35^\circ) = \frac{x}{y}
\]
Solving for \( y \):
\[
y = \frac{x}{\sin(35^\circ)}
\]
We previously found \( x \approx 35.01 \) feet. Now we need to calculate \( \sin(35^\circ) \):
\[
\sin(35^\circ) \approx 0.5736
\]
Thus,
\[
y \approx \frac{35.01}{0.5736} \approx 61.00 \text{ feet}
\]
So the distance \( y \) from Deshaun to the top of the flagpole is approximately **61.00 feet**.
### Summary:
- Height of the flagpole \( x \): **35.01 feet**
- Distance from Deshaun to the top of the flagpole \( y \): **61.00 feet**
1. **Finding the height of the flagpole (\( x \))**:
We can use the tangent function, which relates the angle of elevation to the opposite side (height of the flagpole) and the adjacent side (distance to the base of the flagpole).
The formula is:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Where:
- \( \theta = 35^\circ \)
- Opposite = height of the flagpole \( x \)
- Adjacent = distance to the base of the flagpole = 50 feet
So we rewrite the equation as:
\[
\tan(35^\circ) = \frac{x}{50}
\]
Now solve for \( x \):
\[
x = 50 \tan(35^\circ)
\]
Using a calculator, find \( \tan(35^\circ) \):
\[
\tan(35^\circ) \approx 0.7002
\]
Thus,
\[
x \approx 50 \times 0.7002 \approx 35.01 \text{ feet}
\]
So the height of the flagpole \( x \) is approximately **35.01 feet**.
2. **Finding the distance from Deshaun to the top of the flagpole (\( y \))**:
To find \( y \), we can use the sine function, which relates the angle of elevation to the opposite side (height of the flagpole) and the hypotenuse (distance from Deshaun to the top of the flagpole):
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
So we rewrite the equation as:
\[
\sin(35^\circ) = \frac{x}{y}
\]
Solving for \( y \):
\[
y = \frac{x}{\sin(35^\circ)}
\]
We previously found \( x \approx 35.01 \) feet. Now we need to calculate \( \sin(35^\circ) \):
\[
\sin(35^\circ) \approx 0.5736
\]
Thus,
\[
y \approx \frac{35.01}{0.5736} \approx 61.00 \text{ feet}
\]
So the distance \( y \) from Deshaun to the top of the flagpole is approximately **61.00 feet**.
### Summary:
- Height of the flagpole \( x \): **35.01 feet**
- Distance from Deshaun to the top of the flagpole \( y \): **61.00 feet**