Final answer:
To find the height of the shorter building with an angle of depression of 7° from the top of a taller building 135 ft high, we use the tangent function. After calculating, we find that the shorter building is approximately 110 ft tall.
Step-by-step explanation:
To find the height of the shorter building, we need to apply trigonometric principles. The situation can be visualized as a right triangle, where the taller building is one side, the distance between the buildings is the base, and the line of sight from the top of the taller to the top of the shorter building makes the hypotenuse. The angle of depression is the same as the angle of elevation from the base to the top of the shorter building due to alternate interior angles being equal when a pair of parallel lines are cut by a transversal.
We can use the tangent function here, which is the ratio of the opposite side to the adjacent side in a right-angled triangle. In this case, tangent(7°) = (Height of taller building - Height of shorter building) / Distance between the buildings. We know the height of the taller building (135 ft), the distance between the buildings (200 ft), and the angle (7°).
The calculation looks like this:
-
- tangent(7°) = (135 ft - Height of shorter building) / 200 ft
-
- Height of shorter building = 135 ft - (tangent(7°) × 200 ft)
By using a calculator, we find that tangent(7°) is approximately 0.1228. Therefore:
-
- Height of shorter building = 135 ft - (0.1228 × 200 ft)
-
- Height of shorter building ≈ 135 ft - 24.56 ft
-
- Height of shorter building ≈ 110 ft
The height of the shorter building is approximately 110 ft, to the nearest foot.