From the top of the Metro Building, the angle of elevation of the top of the Gammapro Building is 24° and the angle of depression to the foot of the Gammapro Building is 56°. If the buildings are 45 m apart, determine the height of each building to the nearest meter.
Diagram:
Draw a rectangle with the vert. sides representing the ht. of the Metro Bldg.(h1).and hor. sides are the distance between bldgs.(45m). Draw a diagonal from lower left to upper right and notice that the alternate interior angles are equal(56o).Extend the vert. side on the left and label it h2.
Draw the line of sight from the top of h2 to the top of Metro Bldg. The angle it makes with the hor. is 24o.
Tan56 = h1/45.
h1 = 45*Tan56 = 67 m. = Ht. of Metro bldg.
Tan24 = h2/45.
h2 = 45*Tan24 = 20.0 m.
h1 + h2 = 67 + 20 = 87 m. = Ht. of Gammapro Bldg.
To solve this problem, we will use trigonometric ratios and create two right triangles. Let's denote the height of the Metro Building as h1 and the height of the Gammapro Building as h2.
First, let's draw a diagram to visualize the situation:
C
|\
| \
| \ h2
| \
A|____\____ B
56° 24°
Here, A represents the top of the Metro Building, B represents the top of the Gammapro Building, and C represents the foot of the Gammapro Building.
Now, let's consider the right triangle ABC:
Using the angle of depression, we have: tan(56°) = h1/45 (opposite/adjacent)
Therefore, we can solve for h1 as follows:
h1 = tan(56°) * 45
Now, let's consider the right triangle ABC but from the perspective of the top of the Metro Building:
Using the angle of elevation, we have: tan(24°) = h2/45 (opposite/adjacent)
Therefore, we can solve for h2 as follows:
h2 = tan(24°) * 45
Now, let's calculate the values:
h1 = tan(56°) * 45
h2 = tan(24°) * 45
Using a calculator, we find:
h1 ≈ 71.5 m
h2 ≈ 23.5 m
Therefore, the height of the Metro Building is approximately 71.5 meters, and the height of the Gammapro Building is approximately 23.5 meters.