Akbar and birbal setout at the sametime to walk towards each other respectively from agra and banaras 144 km apart. Akabr walks at the constant speed of 8 km/h. while birbal walks 4 km in the first hour 5 km in the second hour 6 km in the third hour and so on.Then the akbar and birbal will meet.

Options

A. in 6h
B. In 8h
C. midway between Agra and banaras
D. 80 km away fom banras

Answer

To solve the problem of Akbar and Birbal walking towards each other from Agra and Banaras, we will follow these steps:

### Step 1: Understand the Speeds
- Akbar walks at a constant speed of 8 km/h.
- Birbal's speed increases each hour:
- 4 km in the first hour
- 5 km in the second hour
- 6 km in the third hour
- and so on.

### Step 2: Calculate Birbal's Speed for Each Hour
- In the first hour, Birbal walks 4 km.
- In the second hour, he walks 5 km.
- In the third hour, he walks 6 km.
- In general, in the nth hour, he walks (n + 3) km.

### Step 3: Calculate the Total Distance Covered by Both
- The total distance between Agra and Banaras is 144 km.
- We need to find out how many hours it takes for Akbar and Birbal to meet.

### Step 4: Calculate the Distance Covered Each Hour
- In the first hour, the total distance covered by both is:
- Akbar: 8 km
- Birbal: 4 km
- Total: 8 km + 4 km = 12 km

- In the second hour:
- Akbar: 8 km
- Birbal: 5 km
- Total: 8 km + 5 km = 13 km

- In the third hour:
- Akbar: 8 km
- Birbal: 6 km
- Total: 8 km + 6 km = 14 km

- Continuing this pattern, we can see that the total distance covered by both increases by 1 km each hour.

### Step 5: Formulate the Total Distance Equation
- The distance covered in n hours can be expressed as:
- Total distance = 12 km (1st hour) + 13 km (2nd hour) + 14 km (3rd hour) + ... + (11 + n) km (nth hour)

- This is an arithmetic series where the first term (a) is 12 km and the last term (l) is (11 + n) km.

### Step 6: Use the Arithmetic Series Formula
- The sum of the first n terms of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (a + l) \]

- We need to find n such that:
\[ S_n = 144 \text{ km} \]

### Step 7: Set Up the Equation
- Using the formula:
\[ S_n = \frac{n}{2} \times (12 + (11 + n)) = 144 \]
\[ \frac{n}{2} \times (n + 23) = 144 \]
\[ n(n + 23) = 288 \]
\[ n^2 + 23n - 288 = 0 \]

### Step 8: Solve the Quadratic Equation
- Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ n = \frac{-23 \pm \sqrt{23^2 + 4 \times 288}}{2} \]
\[ n = \frac{-23 \pm \sqrt{529 + 1152}}{2} \]
\[ n = \frac{-23 \pm \sqrt{1681}}{2} \]
\[ n = \frac{-23 \pm 41}{2} \]

- This gives us two possible values for n:
- \( n = 9 \) (valid)
- \( n = -32 \) (not valid)

### Step 9: Calculate the Distance Covered by Akbar
- If n = 9, the distance covered by Akbar is:
\[ \text{Distance} = 8 \text{ km/h} \times 9 \text{ hours} = 72 \text{ km} \]

### Step 10: Conclusion
- Therefore, Akbar and Birbal will meet after 9 hours, and the distance covered by Akbar will be 72 km.