How to Solve a Cyclist's Speed and Time Puzzle with Quadratic Equations

Have you ever come across a problem that challenges your problem-solving skills and tests your understanding of mathematical concepts? Well, in this article, we will tackle a classic speed and time puzzle involving a cyclist. This problem involves the use of quadratic equations to find the solution, making it not only an interesting challenge but also a practical application of advanced mathematics.

Understanding the Problem

The problem at hand is as follows:

If a cyclist had gone 3 km/h faster he would have taken 1 hour and 3 minutes less to ride 80 km. How long did it take him originally?

To solve this, we need to use some fundamental principles of distance, speed, and time, as well as a bit of algebra.

Setting Up the Equation

We will denote the original speed of the cyclist as v km/h. The time taken to ride 80 km at this speed can be expressed as:

Timeoriginal 80 / v hours

If the cyclist had gone 3 km/h faster, his speed would be (v 3) km/h, and the time taken at this new speed would be:

Timenew 80 / (v 3) hours

According to the problem, the difference in time between the two scenarios is 1 hour and 3 minutes. We need to convert 1 hour and 3 minutes into hours:

1 hour 3 minutes 1 3/60 1.05 hours

Setting up the equation based on the information given:

80 / v - 80 / (v 3) 1.05

Solving the Equation

To solve this, we find a common denominator:

This simplifies to:

240 / (v(v 3)) 1.05

Next, we cross-multiply:

240 1.05 v(v 3)

Expanding the right side:

240 1.05 v2 3.15 v

Rearranging the equation gives us:

1.05 v2 - 3.15 v - 240 0

To simplify calculations, multiply the entire equation by 100 to eliminate the decimal:

105 v2 - 315 v - 24000 0

Now we can apply the quadratic formula:

v (-b ± √(b2 - 4ac)) / (2a)

where a 105, b -315, and c -24000.

Calculating the Discriminant

The discriminant is:

b2 - 4ac 3152 - 4 * 105 * -24000

99225 10080000 10179225

Now take the square root:

√(10179225) ≈ 3193.02

Now we can find v:

v (-(-315) ± 3193.02) / (2 * 105)

Calculating the two possible values for v:

Using the positive root:

v (315 3193.02) / 210 ≈ 13.72 km/h

Using the negative root would give a negative speed, so we discard it.

Finding the Original Time

Now we can find the time it took the cyclist at this speed:

Time 80 / v 80 / 13.72 ≈ 5.83 hours 5 hours and 50 minutes

Conclusion

The time it took the cyclist originally is 5 hours and 50 minutes. This solution not only demonstrates the use of quadratic equations but also provides a practical scenario that one might encounter in real-life situations involving distance, speed, and time.