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? Round to the nearest hundredth place (2 decimal places).(2 points)
x=
feet and y=
meters.
What is the flagpole’s height, x, and the distance between Deshaun and the top of the flagpole, y? Round to the nearest hundredth place (2 decimal places).(2 points)
x=
feet and y=
meters.
Answers
GPT-4o mini
To determine the height of the flagpole (\(x\)) and the distance from Deshaun to the top of the flagpole (\(y\)), we can use trigonometric relationships.
### 1. **Finding the height of the flagpole (\(x\))**
We can use the tangent of the angle of elevation, which is given by:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case, the angle \(\theta\) is 35°, the opposite side is the height of the flagpole (\(x\)), and the adjacent side is the distance from Deshaun to the base of the flagpole (50 feet).
Thus, we can set up the equation:
\[
\tan(35°) = \frac{x}{50}
\]
Next, we can solve for \(x\):
\[
x = 50 \cdot \tan(35°)
\]
Using a calculator, we find:
\[
\tan(35°) \approx 0.7002 \implies x \approx 50 \cdot 0.7002 \approx 35.01 \text{ feet}
\]
### 2. **Finding the distance to the top of the flagpole (\(y\))**
To find the distance \(y\) from Deshaun to the top of the flagpole, we can use the secant function, or apply the Pythagorean theorem:
\[
y = \sqrt{(50)^2 + (x)^2}
\]
Substituting \(x\) gives:
\[
y = \sqrt{(50)^2 + (35.01)^2}
\]
Calculating this:
\[
y = \sqrt{2500 + 1225.7001} \approx \sqrt{3725.7001} \approx 61.0 \text{ feet}
\]
### 3. **Convert \(y\) to meters**
To convert feet to meters, we use the conversion factor \(1\, \text{foot} \approx 0.3048\, \text{meters}\):
\[
y \approx 61.0 \text{ feet} \times 0.3048 \approx 18.59 \text{ meters}
\]
### Final Answers
Thus, rounding to the nearest hundredth:
- The height of the flagpole (\(x\)) is approximately **35.01 feet**.
- The distance from Deshaun to the top of the flagpole (\(y\)) is approximately **18.59 meters**.
Final results:
\[
x \approx 35.01 \text{ feet}, \quad y \approx 18.59 \text{ meters}
\]
### 1. **Finding the height of the flagpole (\(x\))**
We can use the tangent of the angle of elevation, which is given by:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case, the angle \(\theta\) is 35°, the opposite side is the height of the flagpole (\(x\)), and the adjacent side is the distance from Deshaun to the base of the flagpole (50 feet).
Thus, we can set up the equation:
\[
\tan(35°) = \frac{x}{50}
\]
Next, we can solve for \(x\):
\[
x = 50 \cdot \tan(35°)
\]
Using a calculator, we find:
\[
\tan(35°) \approx 0.7002 \implies x \approx 50 \cdot 0.7002 \approx 35.01 \text{ feet}
\]
### 2. **Finding the distance to the top of the flagpole (\(y\))**
To find the distance \(y\) from Deshaun to the top of the flagpole, we can use the secant function, or apply the Pythagorean theorem:
\[
y = \sqrt{(50)^2 + (x)^2}
\]
Substituting \(x\) gives:
\[
y = \sqrt{(50)^2 + (35.01)^2}
\]
Calculating this:
\[
y = \sqrt{2500 + 1225.7001} \approx \sqrt{3725.7001} \approx 61.0 \text{ feet}
\]
### 3. **Convert \(y\) to meters**
To convert feet to meters, we use the conversion factor \(1\, \text{foot} \approx 0.3048\, \text{meters}\):
\[
y \approx 61.0 \text{ feet} \times 0.3048 \approx 18.59 \text{ meters}
\]
### Final Answers
Thus, rounding to the nearest hundredth:
- The height of the flagpole (\(x\)) is approximately **35.01 feet**.
- The distance from Deshaun to the top of the flagpole (\(y\)) is approximately **18.59 meters**.
Final results:
\[
x \approx 35.01 \text{ feet}, \quad y \approx 18.59 \text{ meters}
\]