SOLUTION: A car travels one lap around a 1 mile track at an average speed of 30 miles per hour. At what average speed must the car travel on the second lap so that the average speed for both

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Question 556337: A car travels one lap around a 1 mile track at an average speed of 30 miles per hour. At what average speed must the car travel on the second lap so that the average speed for both laps is 50 miles per hour.
Found 2 solutions by Alan3354, bucky:
Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website!
A car travels one lap around a 1 mile track at an average speed of 30 miles per hour. At what average speed must the car travel on the second lap so that the average speed for both laps is 50 miles per hour.
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2 mi at 50 mi/hr takes 1/25 hr
1 mi at 30 mi/hr takes 1/30 hr
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He has to go 1 mile in (1/25 - 1/30) hr = 1/150 hr
His speed needed is 150 mi/hr
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Check:
Avg speed for a round trip, or of 2 equal distances is
2*30*150/(30 + 150) = 9000/180
= 50 mi/hr

Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website!
You have to be careful with problems like this one. You can't just say that in order to average 50 mph you can do 1 lap at 30 mph and the second lap at 70 mph because you know that the average of 30 and 70 is (30 + 70)/2 = 100/2 = 50 mph.
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In order to average 50 mph for the two laps of 1 mile each (total distance of 2 miles) you need to determine how much time it would take to cover the total 2 mile distance. Use the equation:
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D = R*T
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where D represents the total distance, R is the rate in mph, and T is the time in hours. Substitute 2 miles for the total distance and 50 mph for the average rate you want to get. Then solve for T as follows:
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2 = 50*T
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Divide both sides by 50 and you get:
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2/50 = T
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Note that on the left side the numerator and the denominator are both divisible by 2. So the fraction on the left side reduces to 1/25 and the equation for the time you have to complete the two laps is 1/25 of an hour.
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Now you can look at how long it took to complete the first lap at a rate of 30 mph. The total distance was 1 mile and the rate was 30 mph. The time it took to make this single lap can be found from the same equation. Start with:
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D = R*T
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Substitute 1 mile for D and 30 mph for R. Then solve for T as below:
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1 = 30*T
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Divide both sides by 30 and you have:
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T = 1/30
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So you complete the first lap in 1/30 hour.
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Now since to average 50 mph you must complete the two laps in 1/25 hour and you used 1/30 of an hour to complete this first lap, you can subtract 1/30 from 1/25 and determine the time you have left to complete the second lap.
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So find:
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1/25 - 1/30 =?
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Let's convert these two to decimals to make it easier to find the difference between them. When you divide 25 into 1 you get 0.0400000 hr and when you divide 30 into 1 you get 0.0333333 hr. So you now can find the time to complete the second lap by subtracting as follows:
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0.0400000 - 0.0333333 = 0.0066666 hr
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So the second 1 mile lap must be completed in a time of 0.0066666 hr. Substitute these two values into the equation:
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D = R*T
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and you get:
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1 = R*0.0066666
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You can solve for the rate in mph by dividing both sides of this equation by 0.0066666 to get:
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1/0.0066666 = R
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Doing the division on the left side results in:
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150 mph = R
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Surprise! You have to drive the second 1-mile lap at 150 mph in order to average 50 mph for the two laps.
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Hope this helps you to understand the problem a little better. And you can see why I warned you not to simply figure that covering 1 lap at 30 mph and the second lap at 70 mph would give you an average speed of 50 mph.
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