Final answer:
The height of the taller building can be found using trigonometry by calculating the elevation and depression angles' impact on the height from Jane's building roof. Assuming Jane's 39-story building is 156 meters tall, the taller building's height would be approximately 163 meters when considering the angles provided and the distance between the buildings.
Step-by-step explanation:
The question involves finding the height of a taller building given the angles of elevation and depression from a point on a nearby building. We can apply trigonometric principles to solve this using the angles and the distance between the two buildings, which is provided as 21 meters. To find the height of the taller building, we will calculate separately the height difference based on the angle of elevation and the height loss based on the angle of depression, and then add these to the height of Jane's building.
Let's use the notation h for the height of the taller building, d for the horizontal distance between the buildings, which is 21 meters, a for the angle of elevation (37 degrees), and b for the angle of depression (24 degrees). Using the tangent function, which is opposite over adjacent in a right triangle, we can write:
tan(a) = height difference / d and tan(b) = height loss / d.
Rearranging these equations, we get height difference = d * tan(a) and height loss = d * tan(b). If x is the height of Jane's building, we can represent h as:
h = x + height difference - height loss.
The height of a single story is approximated as the height of two adult humans, roughly 2 meters each, thus around 4 meters per story. Assuming Jane's building is similar to the 39-story example, we estimate it to be about 156 meters tall (x ~ 4m * 39 stories).
Calculating the height difference and height loss using trigonometry:
Height difference = 21 meters * tan(37 degrees) ≈ 21 meters * 0.7535 ≈ 15.8 meters.
Height loss = 21 meters * tan(24 degrees) ≈ 21 meters * 0.4452 ≈ 9.3 meters.
The estimated height of the taller building would therefore be:
h ≈ 156 meters + 15.8 meters - 9.3 meters ≈ 162.5 meters, rounded to the nearest meter, which is 163 meters.