To solve this problem, let's break it down step-by-step:
1. Understanding the Problem:
- There are two cyclists who are 90 miles apart and start riding toward each other at the same time.
- One cyclist is twice as fast as the other.
- They meet after 3 hours.
- We need to find the speed of the faster cyclist.
2. Define Variables:
- Let's denote the speed of the slower cyclist as [tex]\( x \)[/tex] miles per hour (mi/h).
- Given that the faster cyclist rides at twice the speed of the slower cyclist, the speed of the faster cyclist is [tex]\( 2x \)[/tex] mi/h.
3. Combine their speeds:
- When moving toward each other, their combined speed is the sum of their individual speeds.
- Combined speed: [tex]\( x + 2x = 3x \)[/tex] mi/h.
4. Distance and Time Relationship:
- We know the distance between them is 90 miles.
- They meet after 3 hours, so the total distance covered by both cyclists in 3 hours is 90 miles.
- Using the formula for distance covered: [tex]\( \text{Distance} = \text{Speed} \times \text{Time} \)[/tex]
5. Calculate the Combined Speed:
- Since distance = speed * time, we have:
[tex]\[
90 \text{ miles} = (3x \text{ mi/h}) \times 3 \text{ hours}
\][/tex]
- Simplifying this equation:
[tex]\[
90 \text{ miles} = 9x \text{ miles}
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{90}{9} = 10 \, \text{mi/h}
\][/tex]
6. Determine the Speed of the Faster Cyclist:
- The speed of the slower cyclist is [tex]\( x = 10 \)[/tex] mi/h.
- Therefore, the speed of the faster cyclist is:
[tex]\[
2x = 2 \times 10 \, \text{mi/h} = 20 \, \text{mi/h}
\][/tex]
The speed of the faster cyclist is 20 miles per hour (mi/h).
