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Question

Difficulty: HardRate, Time, and Distance Problems

Two cyclists, Cyclist A and Cyclist B, depart simultaneously from Town X and Town Y, respectively, traveling toward each other along a straight 150-mile path. Cyclist A travels at a constant speed of 18 miles per hour throughout the journey. Cyclist B initially travels at a constant speed of 30 miles per hour. After traveling for 2 hours, Cyclist B encounters a steep incline and decreases speed by 40 percent, maintaining this reduced speed for the remainder of the trip. How many hours after their departure will Cyclist A and Cyclist B meet?

Answer: 3.5 hours

Answer

The total time elapsed from departure until Cyclist A and Cyclist B meet is 3.5 hours.
In the first 2 hours, Cyclist A covers 18×2=3618 \times 2 = 36 miles and Cyclist B covers 30×2=6030 \times 2 = 60 miles, bringing their combined distance to 96 miles and leaving 54 miles remaining. Cyclist B's speed then decreases by 40% to 18 mph (30×0.630 \times 0.6). Moving toward each other, their combined relative rate becomes 18+18=3618 + 18 = 36 mph. Dividing the remaining 54 miles by 36 mph yields 1.5 hours for the second phase. Adding the initial 2 hours gives a total time of 3.5 hours.

Step-by-Step Solution

1
Calculate cumulative distance traveled by both cyclists during the first 2 hours
Cyclist A travels 36 miles; Cyclist B travels 60 miles; Total = 96 miles
Both cyclists travel for 2 full hours at their initial constant speeds.
2
Determine the distance separating the cyclists at t = 2 hours
150 - 96 = 54 miles remaining
Subtract the combined distance covered from the total initial separation of 150 miles.
3
Calculate Cyclist B's new speed after the 40% decrease
30 * 0.60 = 18 mph
A 40% reduction means retaining 60% of the original speed of 30 mph.
4
Calculate relative speed of approach after 2 hours
18 + 18 = 36 mph
When two objects move toward each other, their relative speed is the sum of their individual speeds.
5
Compute the time to cover the remaining distance
54 / 36 = 1.5 hours
Time equals distance divided by relative speed.
6
Sum the time segments to find the total elapsed time
2 + 1.5 = 3.5 hours
The trip consists of an initial 2-hour phase plus an additional 1.5-hour phase.

Key Concept

Relative Speed and Piecewise Motion in Converging Rate Problems
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Topics

Rate, Time, and Distance Problems

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