To determine the height of the building when Amari is standing 50 feet away from its base, and the angle of elevation to the top of the building is [tex]\( 60^\circ \)[/tex], we can use trigonometry, specifically the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side (the height [tex]\( h \)[/tex] of the building in this case) to the adjacent side (the distance [tex]\( d \)[/tex] from Amari to the base).
The steps to solve the problem are as follows:
1. Identify the known values:
- Distance from Amari to the building's base, [tex]\( d = 50 \)[/tex] feet.
- Angle of elevation, [tex]\( \theta = 60^\circ \)[/tex].
2. Use the tangent function:
The tangent of the angle of elevation is given by:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
\][/tex]
3. Plug in the known values:
[tex]\[
\tan(60^\circ) = \frac{h}{50}
\][/tex]
4. Use the value of [tex]\(\tan(60^\circ)\)[/tex]:
We know that [tex]\(\tan(60^\circ) = \sqrt{3}\)[/tex].
5. Set up the equation:
[tex]\[
\sqrt{3} = \frac{h}{50}
\][/tex]
6. Solve for the height [tex]\( h \)[/tex]:
Multiply both sides of the equation by 50:
[tex]\[
h = 50 \times \sqrt{3}
\][/tex]
7. Calculate the numeric value:
[tex]\[
\sqrt{3} \approx 1.732
\][/tex]
[tex]\[
h = 50 \times 1.732 = 86.602 \text{ feet}
\][/tex]
Therefore, the height of the building is approximately [tex]\( 86.60 \)[/tex] feet, matching one of our multiple choice options more concisely:
[tex]\[ 50 \sqrt{3} \text{ feet} \][/tex]
So, the correct answer is [tex]\( 50\sqrt{3} \text{ \ feet} \)[/tex].
