A spherical balloon of radius 10 feet is in the open air. If angle of elevation of centre of the balloon from a point on the ground is `45^(@)` and spherical balloon subtens on angle of `45^(@)` on that point then how high is the centre of the balloon from the ground ?

Options

A. 10 feet
B. 15 feet
C. `20sqrt(2)` feet
D. `10sqrt(2)` feet

Answer

To solve the problem step by step, we will analyze the situation involving the spherical balloon and apply trigonometric principles.

### Step-by-Step Solution:

1. **Understanding the Setup:**
- We have a spherical balloon with a radius of 10 feet.
- The angle of elevation from a point on the ground to the center of the balloon is \(45^\circ\).
- The balloon subtends an angle of \(45^\circ\) at the same point on the ground.

2. **Identifying Key Points:**
- Let \(O\) be the center of the balloon.
- Let \(L\) be the point directly below the center of the balloon on the ground.
- Let \(P\) be the point on the ground from which the angle of elevation is measured.

3. **Analyzing Triangle \(OPL\):**
- In triangle \(OPL\), we know:
- \(\angle PLO = 45^\circ\) (angle of elevation)
- \(OL\) is the height we need to find (the height of the center of the balloon from the ground).
- The side \(PO\) is the hypotenuse of triangle \(OPL\).

4. **Using Trigonometry in Triangle \(OPL\):**
- We can use the sine function:
\[ \sin(45^\circ) = \frac{OL}{PO} \]
- Since \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\), we have:
\[ \frac{1}{\sqrt{2}} = \frac{OL}{PO} \]
- Rearranging gives:
\[ PO = \sqrt{2} \cdot OL \tag{1} \]

5. **Analyzing Triangle \(PAO\):**
- In triangle \(PAO\), where \(A\) is the point where the radius meets the ground:
- \(AO = 10\) feet (the radius of the balloon).
- Again, we can use the sine function:
\[ \sin(45^\circ) = \frac{AO}{PO} \]
- Substituting the known values:
\[ \frac{1}{\sqrt{2}} = \frac{10}{PO} \]
- Rearranging gives:
\[ PO = 10\sqrt{2} \tag{2} \]

6. **Equating the Two Expressions for \(PO\):**
- From (1) and (2), we have:
\[ \sqrt{2} \cdot OL = 10\sqrt{2} \]
- Dividing both sides by \(\sqrt{2}\):
\[ OL = 10 \text{ feet} \]

7. **Conclusion:**
- The height of the center of the balloon from the ground is \(10\) feet.