Context: Abhi and Gita set out at the same time respectively from points P and Q, 81 kms apart towards Q and P, respectively. Abhi walks uniformly at the rate of 4 km/h for his entire journey. Gita also starts walking at the speed of 4 km/h but she increases her speed by 0.25 km/h at the end of every hour. How many hours will Abhi to reach point Q after meeting Gita?

Answer question

Let's solve the problem step by step:

1. **Initial Setup**: Abhi and Gita start from points P and Q respectively, and these points are 81 km apart. Both start at the same time towards each other.

2. **Abhi's Speed**: Abhi walks at a constant speed of 4 km/h.

3. **Gita's Speed**: Gita starts walking at 4 km/h, and she increases her speed by 0.25 km/h at the end of each hour.

### Step 1: Find the time and position of meeting.

We will calculate when and where they meet.

- Let \( t \) be the time (in hours) at which they meet.
  
- Abhi's distance covered by the time of meeting is \( 4t \) km.
  
- Gita's speed changes each hour: 
  - 1st hour: 4 km/h
  - 2nd hour: 4.25 km/h
  - 3rd hour: 4.5 km/h
  - And so on...

- Gita's distance covered in time \( t \) is the sum of the series of distances covered per hour.

Gita's distance after \( t \) hours is given by:
\[
\text{Distance} = 4 + (4 + 0.25) + (4 + 2 \times 0.25) + \ldots + (4 + (t-1) \times 0.25)
\]

Simplifying, this is:
\[
\text{Distance} = t \cdot 4 + 0.25 \cdot (0 + 1 + 2 + \ldots + (t-1))
\]

The sum of the series \(0 + 1 + 2 + \ldots + (t-1)\) is \(\frac{(t-1) \cdot t}{2}\). So, Gita's distance becomes:
\[
\text{Distance} = 4t + 0.25 \cdot \frac{(t-1) \cdot t}{2}
\]

Now, since they meet when their combined distances are 81 km:
\[
4t + \left(4t + 0.25 \cdot \frac{(t-1) \cdot t}{2}\right) = 81
\]

Simplifying this equation:
\[
8t + 0.125 \cdot (t-1) \cdot t = 81
\]

Let's solve this equation to find \( t \).The possible values for the time \( t \) when they meet are \(-72\) hours and \(9\) hours. Since time cannot be negative, the valid solution is \( t = 9 \) hours.

### Step 2: Find the time for Abhi to reach point Q after meeting Gita.

Since Abhi meets Gita after 9 hours, he still needs to cover the remaining distance to point Q:

- Abhi's total journey from P to Q is 81 km.
- By the time they meet, Abhi has already covered \( 4 \times 9 = 36 \) km.
- The remaining distance for Abhi is \( 81 - 36 = 45 \) km.

Abhi walks at a constant speed of 4 km/h. So, the time required for Abhi to cover the remaining 45 km is:

\[
\text{Time} = \frac{45 \text{ km}}{4 \text{ km/h}} = 11.25 \text{ hours}
\]

Therefore, after meeting Gita, Abhi will take **11.25 hours** to reach point Q.

---

Would you like more details or have any questions?

Here are some related questions:

1. How far has Gita walked by the time they meet?
2. What is Gita's speed at the moment they meet?
3. How much distance does Gita cover in the first 5 hours?
4. How long would it take Gita to reach point P?
5. What if Gita increased her speed by 0.5 km/h instead of 0.25 km/h each hour?

**Tip:** For problems involving variable speeds, breaking down the motion into intervals and summing distances covered in each interval is key to finding solutions accurately.

Explore a problem involving Abhi and Gita's journey where they walk towards each other at different speeds. Learn about uniform motion, increasing speed, and solving equations with arithmetic series. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation