When a building under construction was observed from a point P on the ground which isi 120 meters away from its base, the angle of elevation of the top was 30°. Ater is demolition, when it was again observed from the same point in the angle changed to 60°. How much higher was the building raised, from the time it was first observed?
The building was taller after demolition? Very unusual.
Still, ratio of sides in a 30-60-90 triangle is 1:√3:2
So, the height changed from 120/√3 to 120√3
The difference is thus 120(√3-1/√3) = 138.56 m
To find the height difference, we can use trigonometry, specifically the tangent function. Here's how we can approach this problem step by step:
Step 1: Determine the height of the building when it was first observed.
Let's assume the height of the building at that time is h meters. We have a right triangle formed by P, the base of the building, and the top of the building. The angle of elevation is given as 30°.
Using trigonometry, we know that the tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the building (h) and the adjacent side is the distance from the observer to the base of the building (120 meters).
So, we can write the equation tan(30°) = h/120.
To find the value of h, we can rearrange the equation as h = 120 * tan(30°).
Calculating this, we get h ≈ 120 * 0.577 ≈ 69.2 meters.
Therefore, the height of the building when it was first observed is approximately 69.2 meters.
Step 2: Determine the height of the building after demolition.
Similarly, after the demolition, we have a new angle of 60° when observed from point P.
Using the same logic as before, we can write the equation tan(60°) = h'/120, where h' is the new height.
Rearranging the equation, we get h' = 120 * tan(60°).
Calculating this, we get h' ≈ 120 * 1.732 ≈ 207.8 meters.
Therefore, the height of the building after demolition is approximately 207.8 meters.
Step 3: Calculate the height difference.
To determine how much higher the building was raised, we need to find the difference between the two heights.
Height difference = h' - h ≈ 207.8 - 69.2 ≈ 138.6 meters.
Therefore, the building was raised approximately 138.6 meters higher from the time it was first observed.