A balloon is ascending at the constant rate of `9.8 m//sec` at a height of `98 m` above the ground a packet is dropped from it. If time taken by the packet to reach the ground is `(1 + sqrt(3lambda))sec`. Find `lambda`.
(Use `g = 9.8 m//s^(2)`)

Explanation

To solve the problem, we will use the equations of motion to find the value of \( \lambda \).

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Height of the balloon \( h = 98 \, \text{m} \)
- Initial velocity of the packet \( u = 9.8 \, \text{m/s} \) (same as the balloon's ascent rate)
- Acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \) (acting downwards)
- Time taken to reach the ground \( t = 1 + \sqrt{3\lambda} \, \text{s} \)

2. **Set Up the Equation of Motion:**
The equation of motion we will use is:
\[ S = ut + \frac{1}{2} a t^2 \]
Here, the displacement \( S \) is negative since the packet is falling downwards:
\[ S = -98 \, \text{m}, \quad u = 9.8 \, \text{m/s}, \quad a = -g = -9.8 \, \text{m/s}^2 \]

3. **Substituting Values into the Equation:**
Substitute the values into the equation:
\[ -98 = 9.8t - \frac{1}{2} \cdot 9.8 t^2 \]
This simplifies to:
\[ -98 = 9.8t - 4.9t^2 \]
Rearranging gives:
\[ 4.9t^2 - 9.8t - 98 = 0 \]

4. **Multiply through by 2 to eliminate decimals:**
\[ 9.8t^2 - 19.6t - 196 = 0 \]

5. **Use the Quadratic Formula:**
The quadratic formula is given by:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 9.8 \), \( b = -19.6 \), and \( c = -196 \):
\[ t = \frac{19.6 \pm \sqrt{(-19.6)^2 - 4 \cdot 9.8 \cdot (-196)}}{2 \cdot 9.8} \]

6. **Calculate the Discriminant:**
\[ (-19.6)^2 = 384.16 \]
\[ 4 \cdot 9.8 \cdot 196 = 7688.8 \]
\[ \text{Discriminant} = 384.16 + 7688.8 = 8072.96 \]

7. **Calculate \( t \):**
\[ t = \frac{19.6 \pm \sqrt{8072.96}}{19.6} \]
\[ \sqrt{8072.96} \approx 89.87 \]
\[ t = \frac{19.6 \pm 89.87}{19.6} \]
Taking the positive root:
\[ t = \frac{109.47}{19.6} \approx 5.58 \, \text{s} \]

8. **Set Equal to Given Time Expression:**
We know:
\[ t = 1 + \sqrt{3\lambda} \]
Therefore:
\[ 5.58 = 1 + \sqrt{3\lambda} \]
\[ \sqrt{3\lambda} = 5.58 - 1 = 4.58 \]

9. **Square Both Sides:**
\[ 3\lambda = (4.58)^2 \]
\[ 3\lambda = 20.9764 \]
\[ \lambda = \frac{20.9764}{3} \approx 6.9921 \]

10. **Final Calculation for \( \lambda \):**
To find \( \lambda \) as a whole number:
\[ \lambda \approx 7 \]

### Conclusion:
The value of \( \lambda \) is approximately \( 7 \).