Jason B. answered • 04/26/17
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Math Specialist for All Levels
Let's write a function for number of kids, K, in terms of time in seconds, t. I'll write it then explain it.
K(t) = 3 * 2 t / 11
The 3 is the initial number of kids, so at time t = 0, K = 3. We also know that at time t = 11, the number of kids will have to be double what it was at t = 0, then double again at t = 22, etc. This is an exponential growth, which needs an exponential function. Since it's doubling each time, we use 2 as the base of the exponent. Since it only happens every 11 seconds and not every second, we divide t by 11.
Now, we need to solve the equation for K(t) = 88. Let's plug 88 in for K:
88 = 3 * 2 t / 11
88 / 3 = 2 t / 11
t / 11 = log2( 88 / 3 )
t = 11 log2( 88 / 3 )
t = 53.6 seconds
Note: log2( 88 / 3 ) is the same as log( 88 / 3 ) / log( 2 ), which can be put into a calculator
Jason B.
I agree that the number of kids is a discrete function, but I made a different assumption. I assumed that the kids got on the bus one by one throughout each of the 11 second periods (in a sort of quantized continuous exponential function), whereas you assumed that they got on as a new group every 11 seconds. I think my assumption is actually valid and does not contradict the wording of the problem, although it is a less literal interpretation. I stand by it because I also assume that this problem is likely from a math book or other resource, which is likely going to have a numeric answer. I think where the problem creators went wrong is they were trying to be creative but they didn't quite give enough specifics to clearly make it a continuous exponential function.
All that to say, I'm guessing (and I could be wrong) that the answer the creators of this problem are looking for is more likely 53.6 seconds than Technically Never, regardless of whether it's the best answer.
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04/27/17
Kenneth S.
04/26/17