Answer :
To find the height of the building given the provided data, follow these steps:
1. Understand the given information:
- Distance from Amari to the base of the building: [tex]\( 50 \)[/tex] feet.
- Angle of elevation from Amari's position to the top of the building: [tex]\( 60^\circ \)[/tex].
2. Determine the relationship to solve the problem:
- The height of the building and the distance from Amari to the base form a right triangle.
- We can use the tangent function since we know the angle and the adjacent side of the right triangle.
- Recall that [tex]\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)[/tex], where:
- [tex]\(\theta\)[/tex] is the angle of elevation, which is [tex]\( 60^\circ \)[/tex].
- The opposite side is the height of the building.
- The adjacent side is the distance from Amari to the base of the building, [tex]\( 50 \)[/tex] feet.
3. Set up the equation:
[tex]\[ \tan(60^\circ) = \frac{\text{height}}{50 \text{ ft}} \][/tex]
4. Recall the tangent value for [tex]\( 60^\circ \)[/tex]:
[tex]\[ \tan(60^\circ) = \sqrt{3} \][/tex]
5. Plug in the known values:
[tex]\[ \sqrt{3} = \frac{\text{height}}{50 \text{ ft}} \][/tex]
6. Solve for the height:
[tex]\[ \text{height} = 50 \text{ ft} \cdot \sqrt{3} \][/tex]
7. Compute the value:
[tex]\[ 50 \sqrt{3} \text{ ft} \approx 86.60 \text{ ft} \][/tex]
Therefore, the height of the building is [tex]\( 50 \sqrt{3} \)[/tex] feet. After evaluating the given options, we see that the correct answer is:
[tex]\[ \boxed{50 \sqrt{3} \text{ ft}} \][/tex]
1. Understand the given information:
- Distance from Amari to the base of the building: [tex]\( 50 \)[/tex] feet.
- Angle of elevation from Amari's position to the top of the building: [tex]\( 60^\circ \)[/tex].
2. Determine the relationship to solve the problem:
- The height of the building and the distance from Amari to the base form a right triangle.
- We can use the tangent function since we know the angle and the adjacent side of the right triangle.
- Recall that [tex]\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)[/tex], where:
- [tex]\(\theta\)[/tex] is the angle of elevation, which is [tex]\( 60^\circ \)[/tex].
- The opposite side is the height of the building.
- The adjacent side is the distance from Amari to the base of the building, [tex]\( 50 \)[/tex] feet.
3. Set up the equation:
[tex]\[ \tan(60^\circ) = \frac{\text{height}}{50 \text{ ft}} \][/tex]
4. Recall the tangent value for [tex]\( 60^\circ \)[/tex]:
[tex]\[ \tan(60^\circ) = \sqrt{3} \][/tex]
5. Plug in the known values:
[tex]\[ \sqrt{3} = \frac{\text{height}}{50 \text{ ft}} \][/tex]
6. Solve for the height:
[tex]\[ \text{height} = 50 \text{ ft} \cdot \sqrt{3} \][/tex]
7. Compute the value:
[tex]\[ 50 \sqrt{3} \text{ ft} \approx 86.60 \text{ ft} \][/tex]
Therefore, the height of the building is [tex]\( 50 \sqrt{3} \)[/tex] feet. After evaluating the given options, we see that the correct answer is:
[tex]\[ \boxed{50 \sqrt{3} \text{ ft}} \][/tex]
