To solve this problem, we need to use trigonometry. Specifically, we use the tangent function, which relates the angle of elevation to the opposite side (the height of the aerial) and the adjacent side (the distance to the aerial).
Let's break down the problem step by step:
1. Identify the given information:
- The height of the aerial (opposite side): [tex]\( h = 30 \)[/tex] meters.
- The angle of elevation: [tex]\( \theta = 32^\circ \)[/tex].
2. Set up the tangent equation:
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Here, the opposite side is the height of the aerial ([tex]\( h \)[/tex]), and the adjacent side is the distance to the aerial ([tex]\( d \)[/tex]).
[tex]\[
\tan(32^\circ) = \frac{30 \text{ meters}}{d}
\][/tex]
3. Solve for [tex]\( d \)[/tex] (the distance to the aerial):
We need to isolate [tex]\( d \)[/tex] on one side of the equation. Rearrange the equation to solve for [tex]\( d \)[/tex]:
[tex]\[
d = \frac{30 \text{ meters}}{\tan(32^\circ)}
\][/tex]
4. Calculate [tex]\( d \)[/tex]:
Use a calculator to find [tex]\( \tan(32^\circ) \)[/tex]. Make sure your calculator is set to degrees, not radians.
[tex]\[
\tan(32^\circ) \approx 0.6249
\][/tex]
Now, divide the height of the aerial by the tangent of the angle:
[tex]\[
d = \frac{30}{0.6249} \approx 48.02 \text{ meters}
\][/tex]
Therefore, the distance to the aerial is approximately [tex]\( 48.02 \)[/tex] meters.
