A balloon moving in a straight line passes vertically above two points A and B on a horizontal plane 300 ft apart. When above A it has an altitude of `30^(@)` as seen from A. The distance of B it has an altitude of `30^(@)` as seen from A. The distance of B from the point C where it will touch the plane is

Options

A. `150(sqrt3+1)ft`
B. `150ft`
C. `150(3+sqrt3)ft`
D. `300(sqrt3+1)ft`

Answer

To solve the problem step by step, we will use trigonometric relationships in right triangles formed by the balloon's altitude and the horizontal distances between points A, B, and C.

### Step 1: Understand the Configuration
We have two points A and B on a horizontal plane, 300 ft apart. The balloon is at a certain altitude above these points, and we need to find the distance from point B to the point C where the balloon will touch the plane.

### Step 2: Set Up the Triangles
1. **Triangle ABE**: When the balloon is directly above point A, it forms a right triangle with:
- Angle at A (viewing angle from A to the balloon) = 45°
- Opposite side (altitude of balloon above A) = H
- Adjacent side (distance from A to B) = 300 ft

2. **Triangle BEC**: When the balloon is directly above point B, it forms another right triangle with:
- Angle at B (viewing angle from A to the balloon) = 30°
- Opposite side (altitude of balloon above B) = h
- Adjacent side (distance from B to C) = x (the distance we want to find)

### Step 3: Use Trigonometric Ratios
1. **From Triangle ABE**:
- Using the tangent function:
\[ \tan(45^\circ) = \frac{H}{300} \]
- Since \(\tan(45^\circ) = 1\), we have:
\[ H = 300 \text{ ft} \]

2. **From Triangle BEC**:
- Using the tangent function:
\[ \tan(30^\circ) = \frac{h}{x} \]
- We know \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \implies h = \frac{x}{\sqrt{3}} \]

### Step 4: Relate the Heights
Since the balloon is moving in a straight line, we can express the height \(h\) in terms of \(H\):
- The height difference between points A and B is:
\[ H - h = 300 - h \]

### Step 5: Substitute for h
Substituting \(h\) from the previous step:
\[ 300 - \frac{x}{\sqrt{3}} = \text{(the vertical distance from A to B)} \]

### Step 6: Use Similar Triangles
From triangles ABE and BEC being similar:
\[ \frac{H}{300} = \frac{h}{x} \]
Substituting the values we have:
\[ \frac{300}{300} = \frac{\frac{x}{\sqrt{3}}}{x} \]
This simplifies to:
\[ 1 = \frac{1}{\sqrt{3}} \implies x = 300 \sqrt{3} - 300 \]

### Step 7: Solve for x
Now we can express \(x\):
\[ x = 300 \left( \sqrt{3} - 1 \right) \]

### Step 8: Final Calculation
To find the distance from B to C:
\[ x = 150(2\sqrt{3} - 2) = 150(2\sqrt{3} + 2) \]

### Final Answer
The distance from point B to point C is:
\[ x = 150(2 + \sqrt{3}) \text{ feet} \]