To determine the height of the aerial mast, we can use the provided data along with trigonometric principles. Here’s a step-by-step solution:
1. Understand the problem:
- The distance from the foot of the mast to the point where the cable touches the ground is \( 57 \) meters.
- The angle formed by the cable with the ground is \( 32^\circ \).
2. Visualize the situation:
- You can imagine the mast, the cable, and the ground forming a right triangle where:
- The height of the mast is the vertical side.
- The distance from the foot of the mast to the point on the ground where the cable is attached is the horizontal side.
- The cable itself is the hypotenuse.
3. Identify the trigonometric relationship:
- To find the height of the mast, we use the tangent function because it relates the angle of the triangle to the ratio of the opposite side (height of the mast) over the adjacent side (distance on the ground).
- \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
4. Apply the given values:
- In this case:
- \(\theta = 32^\circ\) (the angle)
- \(\text{adjacent} = 57\) meters (the horizontal distance)
5. Set up the equation:
- \(\tan(32^\circ) = \frac{\text{height of the mast}}{57 \text{ meters}}\)
6. Solve for the height of the mast:
- First, calculate \(\tan(32^\circ)\). We find that the tangent of \(32^\circ\) approximately equals \(0.6249\).
- Use the equation: \(0.6249 = \frac{\text{height of the mast}}{57}\)
- Now, solve for the height:
[tex]\[
\text{height of the mast} = 57 \times 0.6249
\][/tex]
7. Perform the multiplication:
- \(\text{height of the mast} = 57 \times 0.6249 \approx 35.62 \text{ meters}\)
Therefore, the height of the mast is approximately [tex]\(35.62\)[/tex] meters.
