Rohan starts to walk 4 km to school at 7:30 am and her brother Eshaan starts at 8:15 am on his bicycle, travelling 12 km/hour faster than Rohan. They reach school at the same time. Represent the given problem in the form of a quadratic equation.

Let's break down the information and formulate it into a quadratic equation step by step.

### Variables and Information:
- Distance to the school: 4 km (for both Rohan and Eshaan).
- Rohan starts at 7:30 AM and walks at a certain speed \( x \) km/h.
- Eshaan starts at 8:15 AM and travels 12 km/h faster than Rohan, so his speed is \( x + 12 \) km/h.
- Both of them reach school at the same time.

### Step 1: Time taken by Rohan to reach school
- Since Rohan starts at 7:30 AM, and his speed is \( x \) km/h, the time Rohan takes to cover the 4 km is:

\[
\text{Time by Rohan} = \frac{4}{x} \, \text{hours}.
\]

### Step 2: Time taken by Eshaan to reach school
- Eshaan starts at 8:15 AM, which is 45 minutes (or 0.75 hours) later than Rohan. His speed is \( x + 12 \) km/h, so the time Eshaan takes to cover the 4 km is:

\[
\text{Time by Eshaan} = \frac{4}{x + 12} \, \text{hours}.
\]

### Step 3: Time difference
- The time difference between their starting times is 45 minutes, or 0.75 hours. Since both of them reach school at the same time, Rohan’s time taken must be 0.75 hours more than Eshaan’s:

\[
\frac{4}{x} = \frac{4}{x + 12} + 0.75.
\]

### Step 4: Form the equation
Now, let's eliminate the fractions by multiplying both sides of the equation by \( x(x + 12) \) to simplify:

\[
4(x + 12) = 4x + 0.75x(x + 12).
\]

Expanding both sides:

\[
4x + 48 = 4x + 0.75x^2 + 9x.
\]

Simplify:

\[
4x + 48 = 4x + 0.75x^2 + 9x.
\]

Cancel out the \( 4x \) from both sides:

\[
48 = 0.75x^2 + 9x.
\]

Multiply through by 4 to eliminate the decimal:

\[
192 = 3x^2 + 36x.
\]

Rearrange into standard quadratic form:

\[
3x^2 + 36x - 192 = 0.
\]

### Final Quadratic Equation:

\[
x^2 + 12x - 64 = 0.
\]

This is the required quadratic equation that represents the problem.

Would you like to see the solution to this equation or need any further details?

Here are 5 related questions for deeper understanding:
1. How do you solve the quadratic equation \(x^2 + 12x - 64 = 0\)?
2. What would happen if Rohan and Eshaan traveled different distances to school?
3. How would the equation change if Rohan walked slower than 4 km/h?
4. Can you generalize this approach for any two travelers starting at different times?
5. How does the speed difference impact the time of travel for Eshaan?

**Tip**: When dealing with problems involving speeds, always account for the time differences first to simplify the creation of equations.

Discover how to formulate and solve a quadratic equation based on the journey of Rohan and Eshaan, who start at different times and speeds but reach the same destination. This problem integrates key algebraic concepts like time, distance, speed, and quadratic equations, ideal for Grades 9-10 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