A cyclist and a jogger start at the same time from the same point. The cyclist travels 15 km/h faster and arrives 2 hours earlier over 90 km. Find both speeds.
Answer (2)
To solve this problem, we need to determine the speeds of the cyclist and the jogger.
Let's introduce some variables to help us organize the information:
Let v be the speed of the jogger in kilometers per hour.
Therefore, the speed of the cyclist will be v + 15 kilometers per hour because the cyclist is 15 km/h faster than the jogger.
We know that the distance traveled by both the jogger and the cyclist is 90 km. We can use the formula for time, which is:
Time = Speed Distance
The jogger's time to cover 90 km is:
Time jogger = v 90
The cyclist's time to cover the same distance is:
Time cyclist = v + 15 90
According to the problem, the cyclist arrives 2 hours earlier than the jogger, so we can set up the equation:
v 90 = v + 15 90 + 2
Let's solve this equation step-by-step:
Multiply through by v ( v + 15 ) to clear the denominators:
90 ( v + 15 ) = 90 v + 2 v ( v + 15 )
Simplify and distribute:
90 v + 1350 = 90 v + 2 v 2 + 30 v
Combine like terms:
1350 = 2 v 2 + 30 v
Rearrange into a standard quadratic equation:
2 v 2 + 30 v − 1350 = 0
Divide every term by 2 to simplify:
v 2 + 15 v − 675 = 0
Solve the quadratic equation using the quadratic formula:
v = 2 a − b ± b 2 − 4 a c
Here, a = 1 , b = 15 , and c = − 675 .
v = 2 × 1 − 15 ± 1 5 2 − 4 × 1 × ( − 675 )
v = 2 − 15 ± 225 + 2700
v = 2 − 15 ± 2925
v = 2 − 15 ± 54.08
Taking the positive solution (as speed cannot be negative):
v = 2 − 15 + 54.08 ≈ 19.54
Therefore, the speed of the jogger is approximately 19.54 km/h, and the speed of the cyclist is:
v + 15 = 19.54 + 15 = 34.54 km/h
Finally, the speeds are approximately:
Jogger: 19.54 km/h
Cyclist: 34.54 km/h
The jogger travels at approximately 19.54 km/h, while the cyclist travels at approximately 34.54 km/h, arriving 2 hours earlier over a distance of 90 km. This was determined by forming an equation based on their speeds and solving the resulting quadratic equation. The relationship between their speeds stems from the difference in time taken to cover the same distance.
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