To determine the height of the building, we can use some basic trigonometry.
1. Identify the Given Values:
- Distance from the building: [tex]\( d = 50 \)[/tex] feet
- Angle of elevation to the top of the building: [tex]\( \theta = 60^\circ \)[/tex]
2. Understand the Relationship:
- We can use the tangent function in trigonometry, which relates the angle of elevation ([tex]\(\theta\)[/tex]), the height of the building ([tex]\(h\)[/tex]), and the distance from the building ([tex]\(d\)[/tex]).
- The tangent of an angle in a right triangle is defined as the ratio of the opposite side (height of the building) to the adjacent side (distance from the building):
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
\][/tex]
3. Set Up the Equation:
- Substitute the given values into the tangent relationship:
[tex]\[
\tan(60^\circ) = \frac{h}{50}
\][/tex]
4. Solve for the Height ([tex]\(h\)[/tex]):
- We need the value of [tex]\(\tan(60^\circ)\)[/tex]. From trigonometric tables or identities, we know:
[tex]\[
\tan(60^\circ) = \sqrt{3}
\][/tex]
- Substitute this value into the equation:
[tex]\[
\sqrt{3} = \frac{h}{50}
\][/tex]
- Solve for [tex]\(h\)[/tex] by multiplying both sides by 50:
[tex]\[
h = 50 \times \sqrt{3}
\][/tex]
5. Calculate the Height (numerically):
- Numerically, [tex]\(\sqrt{3} \approx 1.732\)[/tex], so:
[tex]\[
h \approx 50 \times 1.732 = 86.60254037844383 \text{ feet}
\][/tex]
Thus, the height of the building is approximately 86.60 feet.
The closest choice from the given options is:
[tex]\[
50 \sqrt{3} \text{ feet}
\][/tex]
