We are given that two office buildings are 51 meters apart and that the height of the taller building is 207 meters. We also know that the angle of depression from the top of the taller building to the top of the shorter one is 15^(∘), and we want to evaluate the height of the shorter building. Let's sketch a diagram of this situation.
Notice that a diagram makes the problem much clearer. Since we are asked to find the height of the shorter building, we should find the difference between these buildings and subtract it from 207. Let's call this difference d.
Next let's recall that the angle of depression and corresponding angle of elevation are congruent. Therefore, the angle of elevation from the top of the shorter building to the top of the taller building is also 15^(∘).
To find the value of d, we can use one of the trigonometric ratios. Let's recall that the tangent of an angle is the ratio of the side opposite this angle to the side adjacent to this angle. Using this definition, we can write an equation for tan 15^(∘).
tan15^(∘)=d/51
Let's solve the above equation.
The difference between the heights of the buildings is approximately 14 meters. Finally, we can evaluate the height of the shorter building by subtracting 14 from the height of the taller building, 207.
207-14=193
The height of the shorter building is approximately 193 meters. Notice that this is an approximation as we used an approximate value to evaluate it.
