Question

The angles of depression of the top and bottom of an 8m tall building from the top of a multi-storied building are 30° and 45°, respectively. What is the height of the multi-storied building and the distance between the two buildings?

• A. \( 4(3 + \sqrt{3}) \) m, \( 4(3 + \sqrt{3}) \) m
• B. \( 2(3 + \sqrt{3}) \) m, \( 2(3 + \sqrt{3}) \) m
• C. \( 4(2 + \sqrt{2}) \) m, \( 4(2 + \sqrt{2}) \) m
• D. \( 2(3 - \sqrt{3}) \) m, \( 4(3 + \sqrt{3}) \) m

Hint:

For angles of depression, use the tangent function and solve the resulting system of equations to find the unknowns like height and distance.

Answer 1

The Correct Option is A

Step 1: Use the tangent formula for angles of depression.
From the problem, we have two angles of depression, 30° and 45°, for the top and bottom of the building. The height of the shorter building is 8m. Using the tangent formula for each angle of depression, we can calculate the distances and heights involved. Let the height of the multi-storied building be \( h \), and the distance between the buildings be \( d \). The tangent of the angles of depression gives us the relationships: \[ \tan(30^\circ) = \frac{h - 8}{d_1} \text{and} \tan(45^\circ) = \frac{h}{d_2} \] where \( d_1 \) and \( d_2 \) are the horizontal distances from the top and bottom of the building, respectively.

Step 2: Solve the system of equations.
Using these equations, we solve for the height of the multi-storied building and the distance. After solving, we find that the correct height is \( 4(3 + \sqrt{3}) \) m and the distance is \( 4(3 + \sqrt{3}) \) m.