Question
Lisa and Xiang are going indoor rock climbing. Lisa has set a goal height that she wants to climb, which is at an angle of elevation of 63°. Xiang has set a goal height that is at an angle of elevation 15° greater than Lisa’s. Both are standing at the edge of the mat 20 ft. from the rock wall. How much higher will Xiang climb than Lisa?(1 point)
Responses
Xiang will climb 39.25 ft. higher than Lisa.
Xiang will climb 39.25 ft. higher than Lisa.
Xiang will climb 5.38 ft. higher than Lisa.
Xiang will climb 5.38 ft. higher than Lisa.
Xiang will climb 54.84 ft. higher than Lisa.
Xiang will climb 54.84 ft. higher than Lisa.
Xiang will climb 94.09 ft. higher than Lisa.
Xiang will climb 94.09 ft. higher than Lisa.
Responses
Xiang will climb 39.25 ft. higher than Lisa.
Xiang will climb 39.25 ft. higher than Lisa.
Xiang will climb 5.38 ft. higher than Lisa.
Xiang will climb 5.38 ft. higher than Lisa.
Xiang will climb 54.84 ft. higher than Lisa.
Xiang will climb 54.84 ft. higher than Lisa.
Xiang will climb 94.09 ft. higher than Lisa.
Xiang will climb 94.09 ft. higher than Lisa.
Answers
GPT-4o mini
To find out how much higher Xiang will climb than Lisa, we first need to calculate the height each of them will climb using the tangent function, which relates angles of elevation to opposite and adjacent sides of a right triangle.
1. **Lisa's Climb:**
- Angle of elevation: \(63^\circ\)
- Distance from the wall: \(20 \text{ ft}\)
- Height climbed by Lisa (\(h_L\)) can be calculated using the formula:
\[
h_L = 20 \cdot \tan(63^\circ)
\]
2. **Xiang's Climb:**
- Angle of elevation: \(63^\circ + 15^\circ = 78^\circ\)
- Height climbed by Xiang (\(h_X\)) can be calculated using the formula:
\[
h_X = 20 \cdot \tan(78^\circ)
\]
Now let's calculate \(h_L\) and \(h_X\):
- **Calculating Lisa's height:**
\[
h_L = 20 \cdot \tan(63^\circ) \approx 20 \cdot 1.9626 \approx 39.25 \text{ ft}
\]
- **Calculating Xiang's height:**
\[
h_X = 20 \cdot \tan(78^\circ) \approx 20 \cdot 5.1445 \approx 102.89 \text{ ft}
\]
3. **Difference in height:**
\[
\text{Difference} = h_X - h_L = 102.89 \text{ ft} - 39.25 \text{ ft} = 63.64 \text{ ft}
\]
None of the given options (39.25 ft, 5.38 ft, 54.84 ft, or 94.09 ft) appear to be accurate based on these calculations. The correct height difference is approximately 63.64 ft, indicating a potential issue with the provided options.
However, the calculation reflects the process clearly, and if required to pick the closest existing answer from the given options, none are correct. Thus, based on the calculations, Xiang will climb significantly higher than Lisa.
1. **Lisa's Climb:**
- Angle of elevation: \(63^\circ\)
- Distance from the wall: \(20 \text{ ft}\)
- Height climbed by Lisa (\(h_L\)) can be calculated using the formula:
\[
h_L = 20 \cdot \tan(63^\circ)
\]
2. **Xiang's Climb:**
- Angle of elevation: \(63^\circ + 15^\circ = 78^\circ\)
- Height climbed by Xiang (\(h_X\)) can be calculated using the formula:
\[
h_X = 20 \cdot \tan(78^\circ)
\]
Now let's calculate \(h_L\) and \(h_X\):
- **Calculating Lisa's height:**
\[
h_L = 20 \cdot \tan(63^\circ) \approx 20 \cdot 1.9626 \approx 39.25 \text{ ft}
\]
- **Calculating Xiang's height:**
\[
h_X = 20 \cdot \tan(78^\circ) \approx 20 \cdot 5.1445 \approx 102.89 \text{ ft}
\]
3. **Difference in height:**
\[
\text{Difference} = h_X - h_L = 102.89 \text{ ft} - 39.25 \text{ ft} = 63.64 \text{ ft}
\]
None of the given options (39.25 ft, 5.38 ft, 54.84 ft, or 94.09 ft) appear to be accurate based on these calculations. The correct height difference is approximately 63.64 ft, indicating a potential issue with the provided options.
However, the calculation reflects the process clearly, and if required to pick the closest existing answer from the given options, none are correct. Thus, based on the calculations, Xiang will climb significantly higher than Lisa.