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.

Answer 1

To determine how much higher Xiang will climb than Lisa, we first need to find the height that each person will reach based on their respective angles of elevation and the distance from the wall, which is 20 ft.

1. **Lisa's height:**
- Angle of elevation = 63°
- Distance from the wall = 20 ft

The height can be calculated using the tangent function:
\[
\text{Height} = \text{Distance} \times \tan(\text{Angle}) = 20 \times \tan(63°)
\]

Using a calculator, find \(\tan(63°)\):
\[
\tan(63°) \approx 1.96261
\]
Therefore, Lisa's height is:
\[
\text{Height}_{\text{Lisa}} = 20 \times 1.96261 \approx 39.25 \text{ ft}
\]

2. **Xiang's height:**
- Angle of elevation = 63° + 15° = 78°
- Distance from the wall = 20 ft

Again, using the tangent function:
\[
\text{Height} = \text{Distance} \times \tan(\text{Angle}) = 20 \times \tan(78°)
\]

Using a calculator, find \(\tan(78°)\):
\[
\tan(78°) \approx 4.70463
\]
Therefore, Xiang's height is:
\[
\text{Height}_{\text{Xiang}} = 20 \times 4.70463 \approx 94.09 \text{ ft}
\]

3. **Difference in height between Xiang and Lisa:**
\[
\text{Height difference} = \text{Height}_{\text{Xiang}} - \text{Height}_{\text{Lisa}} = 94.09 \text{ ft} - 39.25 \text{ ft} \approx 54.84 \text{ ft}
\]

Thus, the correct response is:

**Xiang will climb 54.84 ft. higher than Lisa.**