Two students, Tim and Kate, argue about the answer to a SAT exam question they had both encountered. They were trying to
convince one another that THEY were right!
The question was: If you drive halfway to a town 60 miles away at a speed of 30mph, how fast would you need to drive the rest of the way to have an average speed of 60 mph over the entire trip? Assume that any time lost for acceleration or deceleration is negligible.
You need to travel to the town directly (assume a straight line) from your half way position, and once there, you stop and turn off the engine.
The two students came up with very different answers.
Tim says: "It's 90 mph, you buffoon!"
While Kate replies: "You idiot, it can't be done. It's impossible!"
Kate was correct. It is
impossible to achieve this in the 30 miles remaining. In order to achieve an
average speed of 60 mph, you would need to travel at 90 mph for an hour, and
overshoot your destination by 60 miles.
What if you travel faster to boost the average speed? That doesn't work either. The faster you go, the quicker you will reach your destination and therefore the less time at fast speed you have to average out against your 30 mph stint. Because the distance is constant at 30 miles, the distance=speed X time equation always works against you. The higher you push the speed part, the smaller the time part becomes and therefore in 30 miles you can't travel fast enough for long enough to average the first half of the journey out.
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