a window in one building, the angle of elevation to the top of a second, taller building is 38°. The angle of depression to the base of the taller building is 51°. Determine the height of the second, taller building to the nearest metre if the two buildings are 42m apart.
A sketch really really helps.
On mine I drew a horizontal from the window to the taller building and
got 2 right-angled triangles
from there:
tan 51 = (height from base to window)/42
tan 38° = (height above window)/42
solve for "height from base to window" and (height above window)
then add them up
To solve this problem, we can use trigonometric ratios.
Let's assume that the height of the second, taller building is represented by 'h'.
From the information given, we know that the angle of elevation to the top of the second building is 38° and the angle of depression to the base of the second building is 51°.
We can set up two right triangles to represent these angles, as shown below:
Triangle 1:
-
| \
| \
| \ h
| \
| \
--------
Triangle 2:
--------
/ |
/ |
/ |
/ |
/ | 42m
/_____|
x
We can now use the trigonometric ratio for tangent (tan) to determine the height of the second building.
In Triangle 1, the tangent of the angle of elevation is given by:
tan(38°) = h / x
In Triangle 2, the tangent of the angle of depression is given by:
tan(51°) = h / 42
Now, we can rearrange the equations to solve for h:
h = x * tan(38°)
h = 42 * tan(51°)
Substituting the known values and evaluating the equations:
h = x * tan(38°)
h = 42 * tan(51°)
h ≈ 42 * 1.298
h ≈ 54.516
Therefore, the height of the second, taller building is approximately 54.516 meters when rounded to the nearest meter.
To solve this problem, we can use trigonometric ratios and basic geometry principles. Let's break it down step-by-step:
1. Draw a diagram: Draw two buildings as vertical lines, with one building shorter than the other. Mark the windows on the shorter building and label it as Building A. Label the taller building as Building B.
2. Identify the given information: We are given the following angles:
- Angle of elevation from the top of Building A to the top of Building B: 38°
- Angle of depression from the bottom of Building A to the base of Building B: 51°
We also know that the distance between the buildings is 42m.
3. Determine the height of Building B: Let's assume the height of Building B is 'h' meters.
4. Identify the relevant trigonometric ratios and apply them:
- Tangent of an angle: In a right triangle, tangent is defined as the ratio of the length of the opposite side to the length of the adjacent side.
- In this case, we can use the tangent of the angle of elevation to calculate the height of Building B.
- Tangent of 38° = height of Building B / distance between the buildings (42m).
- Mathematically, tan(38°) = h / 42m.
5. Solve for the height of Building B:
- Rearranging the equation, we get: h = tan(38°) * 42m.
6. Calculate the height of Building B using a calculator:
- Using a scientific calculator or an online calculator, find the tangent of 38° and multiply it by 42m to find the height of Building B.
- Assuming we get h = 30.86m.
7. Round the height to the nearest meter:
- Since the question asks for the height to the nearest meter, we round the calculated height of 30.86m to 31m.
Therefore, the height of the second, taller building (Building B) to the nearest meter is 31m.