From the top of a 150m tall building A, the angle of elevation to the top of the building B is 45 degrees and the angle of depression to the bottom of the building B is 30 degrees. What is the height of the building B? Options are (a) 250m (b) `(450)/(sqrt3m)` (c) `150 sqrt3m` (d) `150 (1+ sqrt3)m`.

Options

A. 250m
B. `(450)/(sqrt3m)`
C. `150 sqrt3m`
D. `150 (1+ sqrt3)m`

Answer

To solve the problem, we will use trigonometric concepts involving angles of elevation and depression. Let's break it down step by step.

### Step 1: Understand the Problem
We have two buildings:
- Building A is 150 meters tall.
- We need to find the height of Building B.

From the top of Building A:
- The angle of elevation to the top of Building B is 45 degrees.
- The angle of depression to the bottom of Building B is 30 degrees.

### Step 2: Set Up the Diagram
1. Draw Building A (150m tall).
2. Label the top of Building A as point P and the bottom as point Q.
3. Draw Building B next to it, labeling the top as point T and the bottom as point S.
4. Mark the angles:
- Angle of elevation (from P to T) = 45 degrees.
- Angle of depression (from P to S) = 30 degrees.

### Step 3: Find the Distance from A to B
Using the angle of depression to find the horizontal distance (let's call it QR) from the base of Building A to the base of Building B (point S).

Using the tangent function for angle of depression:
\[ \tan(30^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{150}{QR} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\). Therefore:
\[ \frac{1}{\sqrt{3}} = \frac{150}{QR} \]
Cross-multiplying gives:
\[ QR = 150 \sqrt{3} \text{ meters} \]

### Step 4: Find the Height of the Top of Building B
Now, we will use the angle of elevation to find the height from point P to point T (let's call this height TS).

Using the tangent function for angle of elevation:
\[ \tan(45^\circ) = \frac{TS}{QR} \]
Since \(\tan(45^\circ) = 1\):
\[ 1 = \frac{TS}{150\sqrt{3}} \]
This implies:
\[ TS = QR = 150\sqrt{3} \text{ meters} \]

### Step 5: Calculate the Total Height of Building B
Now we can find the total height of Building B (height from S to T):
\[ \text{Height of Building B} = SR + TS \]
Where:
- \(SR\) (the height from the ground to the bottom of Building B) is equal to the height of Building A, which is 150 meters.
- \(TS\) is the height we just calculated, which is \(150\sqrt{3}\) meters.

Thus:
\[ \text{Height of Building B} = 150 + 150\sqrt{3} \]

### Step 6: Factor the Expression
We can factor out 150:
\[ \text{Height of Building B} = 150(1 + \sqrt{3}) \text{ meters} \]

### Final Answer
The height of Building B is:
\[ \text{Height of Building B} = 150(1 + \sqrt{3}) \text{ meters} \]