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my average speed across both miles? 90 + 30 = 120. 120 / 2 = 60
Your average speed is (total distance)/(total time)
If you drove from Dallas to Houston and someone asked to you calculate your average speed isn't that how you'd do it? You'd figure out the distance between Houston and Dallas and divide by the time it took you to make the trip.
You wouldn't log your speed over every mile of the trip and add it all together and then average it out.
I still disagree. The MPH speed of the car is simply a rate. It doesn't in any way imply time except when measuring distances travelled over time. The question, as posed, doesn't suggest anything about the time in which the trip takes place. Yes, you'll end up taking longer than 2 minutes and, as a function a full hour's trip, you'd be right that your average speed wasn't 60, but your average rate of travel was. And that's what the problem was asking. 30 + 90 = 120 / 2 = 60 average
When you are calculating the average of rates, you are looking for the harmonic mean and not the arithmetic mean. You got the rate right but not the right mean.
The formula you are looking for is this: 2ab/a+b. 2(30*90)/(30+90)=45
Your average speed is (total distance)/(total time)
If you drove from Dallas to Houston and someone asked to you calculate your average speed isn't that how you'd do it? You'd figure out the distance between Houston and Dallas and divide by the time it took you to make the trip.
You wouldn't log your speed over every mile of the trip and add it all together and then average it out.
If you use the harmonic mean to calculate the average for the second method, you will get the same answer as your first method. Of course, one method is much simpler than the other.
I see how the 90 MPH people can be confused. Sometimes logic is not the way to convince them. They are to be treated gently.
The correct answer was stated, and yes it's impossible to average 60 MPH over the 2 miles. But frankly, as soon as that woman's image was posted, I completely lost interest in the question. Where did that pic come from, who is it, and where else can we see her in media?? If she IS in the car, I would just pull over at the top and never drive down the other side!
Yes but some of the posts mentioned 2 minutes to travel the distance and nowhere in the op's statement that there is a 2 minute time limit to travel the distance I think you all are overthinking a very simple question , you are traveling 2 miles you avg 30 mph for the first mile what speed do you need to travel in the other mile to avg 60 mph for the 2 miles. where the 2 minutes come from is beyond me
Actually he did define a time limit of two minutes when he said average 60 miles per hour. 60 miles per hour is one mile per minute. To travel two miles at an average of 1 mile per minute means the time is limited to no more than 2 minutes or no less than 2 minutes.
The speed (v=distance/time) required to go down hill would be infinity (v=1/0) which is impossible since even the speed of light in free space is limited to 186,282.4 miles per second or 11,176,944 miles per minute.
Bill
Last edited by Bill Dearborn; 10-12-2018 at 05:16 PM.
I've got a coupla math experts in my family...one of them gave me this mathematical answer...
It is impossible, but let's expand the problem a bit:
Let x = your uphill speed in mph and let y = your downhill speed in mph. Keep the supposition that your goal is an average of 60mph over 2 miles. Then y = 30x/(x-30), and the graph would look like this:
Notice a vertical asymptote at x = 30. That was the original impossible scenario. There is also a horizontal asymptote at y = 30 since it would also be impossible if your downhill speed were 30 mph. All points of the graph to the left of the vertical asymptote are meaningless in terms of this problem, but I've highlighted a few interesting points on the right. For example, if you drive 39 mph uphill, then you would need to drive 130 mph downhill to hit your average speed goal of 60 mph. If you could drive 80 mph uphill, then you would need to slow to 48 mph going downhill.
You could write a couple of limit statements:
AND
So, the closer you get to 30 mph going uphill, your downhill speed will have to approach infinity. And as your uphill speed approaches infinity, your downhill speed will approach 30 mph.
Thanks for the interesting problem!! I'm going to use this next semester when we study rational functions.
Yes, the two limit statements summarize things nicely.
Btw, I should credit Gerd Gigerenzer, who wrote about the problem in his book called Risk Savvy (pp. 43-44). Gerd had access to Max Wertheimer's private correspondence. He has copies of Wertheimer's letter and Einstein's response. (See my post #42 in this thread.)
10-12-2018, 08:35 AM
