The angles of depressions of the top and bottom of 8m tall building from the top of a multistoried building are `30^@ and 45^@` respectively. Find the height of multistoried building and the distance between the two buildings.

Answer

To solve the problem, we will follow these steps:

### Step 1: Draw the Diagram
First, we need to visualize the situation. We have two buildings: an 8-meter tall building (let's call it Building A) and a multi-storied building (let's call it Building B). The top of Building B is point C and the bottom is point D. The angles of depression from point C to the top (A) and bottom (B) of Building A are 30° and 45° respectively.

### Step 2: Identify the Angles
From the diagram:
- The angle of depression from C to A is 30°.
- The angle of depression from C to B is 45°.

Since the angles of depression are equal to the angles of elevation from points A and B to point C, we have:
- Angle EAC = 30°
- Angle EBD = 45°

### Step 3: Set Up the Right Triangles
Now, we can set up two right triangles:
1. Triangle AEC (for the angle of depression to the top of Building A).
2. Triangle CBD (for the angle of depression to the bottom of Building A).

Let AE (the height from the top of Building B to the top of Building A) be x, and the height of Building B (from D to C) be h.

### Step 4: Use Trigonometric Ratios
Using the tangent function for both triangles:
1. For triangle AEC:
\[ \tan(30°) = \frac{AE}{EC} \implies \frac{1}{\sqrt{3}} = \frac{x}{d} \implies d = x \sqrt{3} \]

2. For triangle CBD:
\[ \tan(45°) = \frac{CD}{BD} \implies 1 = \frac{x + 8}{d} \implies d = x + 8 \]

### Step 5: Set the Equations Equal
From the two equations for d:
\[ x \sqrt{3} = x + 8 \]

### Step 6: Solve for x
Rearranging gives:
\[ x \sqrt{3} - x = 8 \implies x(\sqrt{3} - 1) = 8 \implies x = \frac{8}{\sqrt{3} - 1} \]

### Step 7: Rationalize the Denominator
To simplify \( x \):
\[ x = \frac{8(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{8(\sqrt{3} + 1)}{3 - 1} = 4(\sqrt{3} + 1) \]

### Step 8: Find the Height of Building B
The height of Building B (h) is:
\[ h = x + 8 = 4(\sqrt{3} + 1) + 8 = 4\sqrt{3} + 12 \]

### Step 9: Find the Distance Between the Buildings
The distance between the two buildings (d) is:
\[ d = x + 8 = 4(\sqrt{3} + 1) + 8 = 4\sqrt{3} + 12 \]

### Final Answers
- Height of the multi-storied building (h) = \( 4\sqrt{3} + 12 \) meters
- Distance between the two buildings (d) = \( 4\sqrt{3} + 12 \) meters