The feet of two building of height 30miter amd 70meter are on the same horizontal line with a hole on the ground.The smaller building being between the taller one and the hole is 200meter the angle of elevation of the top of the taller building from the top of the smaller one is 30digree. Calculate: The distance from the top of the taller building to the top of the smaller one. The distance of the foot of the taller building the hole. The angle of elevation of the top of the taller building from the hole

To solve this problem, we can use trigonometric ratios and the concept of similar triangles. Let's break it down step by step:

Step 1: Draw a diagram
Start by drawing a diagram that represents the given information. Label the heights of the two buildings as 30 meters and 70 meters, and label the distance between the buildings as 200 meters. Place a dot to represent the hole on the ground.

Step 2: Find the distance from the top of the taller building to the top of the smaller one
We can create a right triangle using the top of the taller building, the top of the smaller building, and a point vertically below the top of the smaller building. Since we have the height of the smaller building (30 meters) and the angle of elevation (30 degrees), we can use the tangent ratio:

tan(angle) = opposite/adjacent

We know that the angle is 30 degrees and the opposite side is 30 meters. Let's denote the distance we are looking for as x:

tan(30 degrees) = 30/x

Rearranging the equation, we get:

x = 30/tan(30 degrees)

Using a calculator, we find that x is approximately 51.96 meters. Therefore, the distance from the top of the taller building to the top of the smaller one is approximately 51.96 meters.

Step 3: Find the distance from the foot of the taller building to the hole
To find this distance, we need to use the similar triangles formed by the two buildings. By examining the diagram, we see that the ratio of the heights of the buildings is the same as the ratio of the distances from the foot of the taller building to the hole and from the top of the taller building to the top of the smaller one.

Let's denote the distance we are looking for as y:

70/30 = y/51.96

Simplifying the equation, we get:

y = (70/30) * 51.96

Using a calculator, we find that y is approximately 120.98 meters. Therefore, the distance from the foot of the taller building to the hole is approximately 120.98 meters.

Step 4: Find the angle of elevation of the top of the taller building from the hole
To find this angle, we can use the previously calculated distance (y) and the height of the taller building (70 meters). Again, we can use the tangent ratio:

tan(angle) = opposite/adjacent

We know that the opposite side is 70 meters and the adjacent side is y. Let's denote the angle of elevation as A:

tan(A) = 70/y

Rearranging the equation, we get:

A = tan^(-1)(70/y)

Using a calculator, we find that A is approximately 29.15 degrees. Therefore, the angle of elevation of the top of the taller building from the hole is approximately 29.15 degrees.