Question
You are looking up at a flag that is 25 meters away from you at a 40-degree angle of elevation with the ground. To the nearest meter, what is the height of the flagpole, x, and the distance between you and the top of the flagpole, y? Round to the hundredths place (2 decimal places) as needed.(2 points) x= meters and y= meters.
Answers
GPT-4o mini
To find the height of the flagpole (x) and the distance from you to the top of the flagpole (y), we can use basic trigonometry.
1. **Find the height of the flagpole (x)**:
We can use the tangent of the angle of elevation. The tangent of an angle in a right triangle is the ratio of the opposite side (height of the flagpole) to the adjacent side (distance from you to the base of the flagpole).
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, \(\theta = 40^\circ\), and the adjacent side is 25 meters:
\[
\tan(40^\circ) = \frac{x}{25}
\]
Rearranging this to find x:
\[
x = 25 \cdot \tan(40^\circ)
\]
Now, calculate \(x\):
\[
x \approx 25 \cdot 0.8391 \quad (\text{using } \tan(40^\circ) \approx 0.8391)
\]
\[
x \approx 20.98 \text{ meters}
\]
Rounding to the nearest meter:
\[
x \approx 21 \text{ meters}
\]
2. **Find the distance to the top of the flagpole (y)**:
The distance to the top of the flagpole forms the hypotenuse of the right triangle. We can use the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Rearranging the formula to find the hypotenuse:
\[
\sin(40^\circ) = \frac{x}{y}
\]
\[
y = \frac{x}{\sin(40^\circ)}
\]
Substitute \(x\):
\[
y \approx \frac{20.98}{0.6428} \quad (\text{using } \sin(40^\circ) \approx 0.6428)
\]
Calculate \(y\):
\[
y \approx 32.67 \text{ meters}
\]
Rounding to two decimal places:
\[
y \approx 32.67 \text{ meters}
\]
**Final Answers:**
- \(x \approx 21\) meters
- \(y \approx 32.67\) meters
1. **Find the height of the flagpole (x)**:
We can use the tangent of the angle of elevation. The tangent of an angle in a right triangle is the ratio of the opposite side (height of the flagpole) to the adjacent side (distance from you to the base of the flagpole).
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, \(\theta = 40^\circ\), and the adjacent side is 25 meters:
\[
\tan(40^\circ) = \frac{x}{25}
\]
Rearranging this to find x:
\[
x = 25 \cdot \tan(40^\circ)
\]
Now, calculate \(x\):
\[
x \approx 25 \cdot 0.8391 \quad (\text{using } \tan(40^\circ) \approx 0.8391)
\]
\[
x \approx 20.98 \text{ meters}
\]
Rounding to the nearest meter:
\[
x \approx 21 \text{ meters}
\]
2. **Find the distance to the top of the flagpole (y)**:
The distance to the top of the flagpole forms the hypotenuse of the right triangle. We can use the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Rearranging the formula to find the hypotenuse:
\[
\sin(40^\circ) = \frac{x}{y}
\]
\[
y = \frac{x}{\sin(40^\circ)}
\]
Substitute \(x\):
\[
y \approx \frac{20.98}{0.6428} \quad (\text{using } \sin(40^\circ) \approx 0.6428)
\]
Calculate \(y\):
\[
y \approx 32.67 \text{ meters}
\]
Rounding to two decimal places:
\[
y \approx 32.67 \text{ meters}
\]
**Final Answers:**
- \(x \approx 21\) meters
- \(y \approx 32.67\) meters