Certainly! Let's tackle this problem step-by-step.
1. Understand the given information:
- The distance between the foot of the building and the bus is 20 meters.
- The angle of depression from the top of the building to the bus is 45°.
2. Visualize the problem:
- Imagine you're standing on top of the building looking down. The line of sight from the top to the bus forms the angle of depression with the horizontal line (imagine a line parallel to the ground starting from your eye level).
3. Relate the angle of depression to the angle of elevation:
- The angle of depression from the top of the building to the bus equals the angle of elevation from the bus to the top of the building. Both are 45° because they are alternate interior angles.
4. Set up a right triangle:
- The foot of the building, the bus, and the top of the building form a right triangle.
- The horizontal leg of the triangle is the distance between the foot of the building and the bus (20 meters).
- The vertical leg is the height of the building, which we need to find.
- The angle of elevation (or depression) is 45°.
5. Use trigonometric ratios:
- In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Here, for the 45° angle:
[tex]\[
\tan(45°) = \frac{\text{height of the building}}{\text{distance to the building}}
\][/tex]
6. Substitute the known values:
- [tex]\[
\tan(45°) = \frac{\text{height of the building}}{20}
\][/tex]
7. Recall the value of [tex]\(\tan(45°)\)[/tex]:
- [tex]\[
\tan(45°) = 1
\][/tex]
8. Set up the equation:
- [tex]\[
1 = \frac{\text{height of the building}}{20}
\][/tex]
9. Solve for the height of the building:
- To find the height, isolate the variable:
[tex]\[
\text{height of the building} = 1 \times 20
\][/tex]
[tex]\[
\text{height of the building} = 20 \text{ meters}
\][/tex]
10. Conclusion:
- The height of the building is 20 meters.
Note: All calculations agree with the original trigonometric values and the logic used in solving such right-angle triangle problems.
