Speed Calculation for Raj's Commute: A Real-Life Application of Quadratic Equations

Imagine a student named Raj whose school is 80 kilometers away from his house. One day, Raj was an hour late, leaving for school and deciding to increase his speed to make up for the lost time. This real-life scenario can be solved using a quadratic equation, a fundamental concept in algebra. Let's explore how to find Raj's changed speed through this problem.

Setting Up the Problem

Let's denote the normal travel time from Raj's house to school as T (in hours). At this speed, Raj covers 80 kilometers. So, the normal travel speed can be calculated as:

Normal Speed

Normal speed

80/T km/h.

Increasing Speed for Late Departure

One day, Raj left an hour late, increasing his speed by 4 km/h. We need to find the new speed, which will allow him to reach school at the normal time. Let's break down the steps to solve this problem.

Step 1: Formulate the Equation

Let the increased speed be X 4 km/h. The time taken to travel 80 km with this new speed will be:

Time

80/(X 4) hours.

Since Raj was an hour late, the new travel time will be T - 1 hours. Therefore, we set up the equation as:

80/(X 4) T - 1.

Step 2: Simplify the Equation

We can rewrite the equation as:

80/(X 4) (80/X) - 1.

Multiplying both sides by (X 4)X gives:

8 (X 4)(T - 1).

Expanding the right side:

8 XT - X 4T - 4.

Collecting like terms:

(80 - T)X 4T - 4.

A convenient substitution here is to let X 80/T. Thus:

(80 - T)(80/T) 4T - 4.

Step 3: Solve the Quadratic Equation

Multiplier the left side:

6400/T - 80 4T - 4 0.

Combining terms:

6400 - 80T 4T^2 - 4T 0.

Dividing everything by 4:

T^2 - 20T - 20 0.

Step 4: Apply the Quadratic Formula

The quadratic formula is X [-b plusmn; sqrt(b^2 - 4ac)] / 2a. For the equation T^2 - 20T - 20 0, a 1, b -20, and c -20. So:

T (20 plusmn; sqrt(400 80)) / 2.

Further simplification:

T (20 plusmn; sqrt(480)) / 2.

Since T must be positive:

T (20 sqrt(480)) / 2 5 hours.

Step 5: Find the New Speed

The required increased speed is

80 km/5 4 km/h 16 4 km/h 20 km/h.

Conclusion

Raj's normal travel speed was 16 km/h, and when he increased it by 4 km/h, his new speed became 20 km/h. This solution demonstrates the real-world application of quadratic equations and how they can help solve practical problems in our daily lives. Understanding such mathematical concepts can greatly assist in managing time efficiently, as seen in Raj's commute.