A balloon moving in a straight line passes vertically above two points A and B on a horizontal plane 10ft apart. When above A the balloon has an angle of elevation of `60^(@)` as seen from B. When above B it has an angle of elevation of `45^(@)` as seen from A. The distance of B from the point C where it will touch the plane is

Options

A. `5(sqrt3+1)" ft"`
B. 15 ft
C. `5(3+sqrt3)ft`
D. None of these

Answer

To solve the problem, we will use trigonometric relationships and properties of triangles. Let's break it down step by step.

### Step 1: Understand the Geometry
We have two points A and B on the horizontal plane, which are 10 feet apart. Let the height of the balloon above point A be \( h_A \) and above point B be \( h_B \).

### Step 2: Set Up the Angles
From point B, when the balloon is directly above A, the angle of elevation is \( 60^\circ \). From point A, when the balloon is directly above B, the angle of elevation is \( 45^\circ \).

### Step 3: Use Trigonometric Ratios
Using the tangent function, we can express the heights in terms of the distances:

1. From point B to point A:
\[ \tan(60^\circ) = \frac{h_A}{10} \]
Therefore,
\[ h_A = 10 \tan(60^\circ) = 10 \cdot \sqrt{3} = 10\sqrt{3} \text{ feet} \]

2. From point A to point B:
\[ \tan(45^\circ) = \frac{h_B}{10} \]
Therefore,
\[ h_B = 10 \tan(45^\circ) = 10 \cdot 1 = 10 \text{ feet} \]

### Step 4: Find the Height Difference
Let \( R \) be the point where the balloon will touch the plane. The height of the balloon above point A is \( h_A = 10\sqrt{3} \) and above point B is \( h_B = 10 \). The vertical distance from point B to point R (the point where the balloon touches the ground) is \( h_B \).

### Step 5: Set Up the Equation for the Distance
Let \( x \) be the distance from point B to point R. The height from point B to point R is:
\[ h_B = x \tan(45^\circ) = x \]
So, we have:
\[ 10 = x \quad \Rightarrow \quad x = 10 \text{ feet} \]

### Step 6: Find the Total Distance from A to C
The total distance from A to C (where the balloon will touch the plane) is:
\[ AC = AB + BC = 10 + x = 10 + 10 = 20 \text{ feet} \]

### Final Answer
The distance of B from the point C where it will touch the plane is:
\[ \boxed{10 \text{ feet}} \]