The angle of elevation of the top of the building from the foot of the tower is 30 and the angle of the top of the tower from the foot of the building is 60. If the tower is 50/3 m high, find the height of the building.

To solve this problem, we can use the trigonometric ratios, specifically the tangent of an angle in a right triangle.

Step 1: Let's denote the height of the building as h and the distance between the tower and the building as d.

Step 2: From the given, we know that the angle of elevation of the top of the building from the foot of the tower is 30 degrees. This means that tan(30) = h/d.

Step 3: We also know that the angle of elevation of the top of the tower from the foot of the building is 60 degrees. This means that tan(60) = (h + 50/3) / d.

Step 4: We know that tan(30) = 1/√3 and tan(60) = √3. Substituting these values into the equations from step 2 and 3, we get:

1/√3 = h/d and √3 = (h + 50/3) / d.

Step 5: From the first equation, we can express d as d = √3 * h.

Step 6: Substituting d from step 5 into the second equation, we get √3 = (h + 50/3) / (√3 * h).

Step 7: Simplifying this equation, we get h^2 + 50√3/3 = h.

Step 8: Solving this quadratic equation, we get h = 25/√3 m.

So, the height of the building is 25/√3 m.