2 math questions. Please show your work if you can, I struggle with this, Thanks !! 20pts
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Answers:
Goods train has a speed of 80 km per hour
Express train has a speed of 100 km per hour
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Explanation:
Check out the diagram below to see the table. For now, refer to table 1.
Each row represents a different train.
We have x as the speed of the slower train (the goods train) and x+20 is the speed of the faster train (express train). These speeds are in km/hr or kph.
The y variable represents the number of hours it takes the goods train to arrive at its destination. The equation formed in the first row is 1040 = xy. This is because distance = rate*time.
Let's solve for y to get
1040 = xy
xy = 1040
y = 1040/x
We'll use this equation later.
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The second row of table 1 has x+20 as the speed and y-2-0.6 as the time.
The "-2" represents the fact that the express train is operating for 2 hours less compared to the goods train (because the goods train had a 2 hour head start). Then the "-0.6" indicates we're taking off another 36 minutes, which is equivalent to 36/60 = 0.6 hours.
So overall, the express train is traveling for 2+0.6 = 2.6 fewer hours compared to the goods train. The express train's time value is y-2.6 hours.
With this in mind, we can form this equation for the second row
distance = rate*time
1040 = (x+20)(y-2.6)
We'll now apply substitution to replace y with 1040/x. This is valid because earlier we found that y = 1040/x
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If we apply that substitution and solve for x, we get the following
Let's now use the quadratic formula to finish things off.
Ignore the negative solution because we can't have a negative speed.
The only practical solution here is x = 80
This means the goods train has a speed of 80 km/hr.
The express train's speed must be x+20 = 80+20 = 100 km/hr.
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Now to verify these answers.
If the goods train travels 1040 km and does so at a speed of 80 km/hr, then it travels for 1040/80 = 13 hours. Add this onto 6:00 PM and we arrive at 7:00 AM. This is shown in the first row of table 2 (see below).
If the express train travels 1040 km and its speed is 100 kph, then it travels for 1040/100 = 10.4 hours. This is equivalent to 10 hours, 24 minutes because 0.4 hrs = 0.4*60 = 24 min
If we start at 8:00 PM and elapse 10 hours, then we'll arrive at 6:00 AM. Add on another 24 minutes, and we get to 6:24 AM. Notice how this is 36 minutes before 7:00 AM (because 24+36 = 60). So this confirms that the express train indeed arrives 36 minutes before the goods train does, despite the goods train having that 2 hour head start.
