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Zorluk: Çok zorRate, Time, and Distance Problems

Train X and Train Y depart simultaneously from Station A and Station B, respectively, heading toward each other along a straight, 360-mile track. Train X travels at a constant speed of 40 miles per hour for the first 2 hours, after which its speed increases by 50 percent. Train Y travels at a constant speed of 50 miles per hour for the first hour, stops for 30 minutes due to a signal delay, and then resumes its journey at a constant speed of 70 miles per hour. How many hours after their simultaneous departure will the two trains meet?

Cevap: 3.5 hours

Cevap

3.5 hours
Evaluating distance piecewise up to t=2t = 2 hours shows Train X covers 80 miles and Train Y covers 85 miles, leaving 195 miles between them. After t=2t = 2 hours, their relative rate of convergence is 60+70=13060 + 70 = 130 mph. Dividing 195 miles by 130 mph yields 1.5 additional hours, bringing the total time since departure to 3.5 hours.

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1
Determine distance traveled by Train X in the first 2 hours and its new speed.
Train X distance = 40 mph×2 hours=80 miles40 \text{ mph} \times 2 \text{ hours} = 80 \text{ miles}. New speed = 40×(1+0.50)=60 mph40 \times (1 + 0.50) = 60 \text{ mph}.
Train X increases its speed by 50% after the 2-hour mark.
2
Determine distance traveled by Train Y during the first 2 hours.
In hour 1 (t=0t=0 to t=1t=1): 50 mph×1 h=50 miles50 \text{ mph} \times 1 \text{ h} = 50 \text{ miles}. During delay (t=1t=1 to t=1.5t=1.5): 0 miles0 \text{ miles}. From t=1.5t=1.5 to t=2.0t=2.0: 70 mph×0.5 h=35 miles70 \text{ mph} \times 0.5 \text{ h} = 35 \text{ miles}. Total distance = 50+0+35=85 miles50 + 0 + 35 = 85 \text{ miles}.
Accounting for Train Y's initial leg, 30-minute delay, and new speed up to t=2t = 2 hours.
3
Find the remaining distance separating the two trains at t=2t = 2 hours.
Total distance covered by both trains = 80+85=165 miles80 + 85 = 165 \text{ miles}. Remaining distance = 360−165=195 miles360 - 165 = 195 \text{ miles}.
Subtracting total distance covered by both trains from the track length of 360 miles.
4
Calculate the time required to cover the remaining distance using relative speed.
Relative closing speed for t>2t > 2 is 60+70=130 mph60 + 70 = 130 \text{ mph}. Additional time = 195130=1.5 hours\frac{195}{130} = 1.5 \text{ hours}.
When two objects move toward each other, their rates add together to find the closing rate.
5
Calculate total elapsed time from departure.
Total time = 2.0+1.5=3.5 hours2.0 + 1.5 = 3.5 \text{ hours}.
Combining the initial 2-hour evaluation window with the additional 1.5 hours required to meet.

Anahtar Kavram

Relative speed and piecewise distance calculations for converging vehicles with variable speeds and delays.
Tahmini Süre:3m 0s
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Rate, Time, and Distance Problems

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