To determine the distance between the observer and the foot of the tower, we'll use trigonometric relationships, specifically those involving the tangent function. Here's a detailed, step-by-step solution:
1. Identify the given parameters:
- The angle of elevation ([tex]\(\theta\)[/tex]) is [tex]\(30^\circ\)[/tex].
- The height of the tower (opposite side to the angle) is 50 meters.
2. Understand the trigonometric relationship:
- We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle.
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
3. Substitute the known values into the tangent formula:
- Here, the opposite side is the height of the tower (50 meters), and the adjacent side is the distance from the observer to the foot of the tower, which we'll denote as [tex]\(d\)[/tex].
So, our equation becomes:
[tex]\[
\tan(30^\circ) = \frac{50}{d}
\][/tex]
4. Find the tangent of the given angle:
- The tangent of [tex]\(30^\circ\)[/tex] is a known value:
[tex]\[
\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577
\][/tex]
5. Set up the equation with the known tangent value:
[tex]\[
\frac{1}{\sqrt{3}} = \frac{50}{d}
\][/tex]
6. Solve for [tex]\(d\)[/tex]:
- To isolate [tex]\(d\)[/tex], multiply both sides of the equation by [tex]\(d\)[/tex] and then by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
d \times \frac{1}{\sqrt{3}} = 50
\][/tex]
[tex]\[
d = 50 \sqrt{3}
\][/tex]
7. Calculate the value of [tex]\(d\)[/tex]:
- Approximating [tex]\(\sqrt{3}\)[/tex] to 1.732, we get:
[tex]\[
d = 50 \times 1.732
\][/tex]
[tex]\[
d \approx 86.6 \text{ meters}
\][/tex]
Conclusion:
The observer is approximately 86.6 meters away from the foot of the tower.
