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Let’s imagine that the number of rabbits in a field doubles each month. If we start with 6 rabbits, how many rabbits would be in the field after 10 weeks, given that 4 weeks = 1 month?

I would think the answer would be $$6 * 2^{2.5}$$ which equals a non-integer answer, but a colleague of mine argues that the answer would be $$(6 * 2^2)+ (6 * 2^2 * \frac{1}{2})$$ which is an integer answer but I think is incorrect. Can someone explain the correct way of thinking about this problem and which answer is correct, or if even neither of the solutions are correct?

I'm not asking how to calculate non-integer exponents, but rather when their use is appropriate.

S.Kumar
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  • I agree with yours rather than his. It seems that to explain his answer he found the number of rabbits after two months ($6\cdot 2^2$) and then somehow concluded that with that amount of rabbits, after two weeks the amount will increase by $50%$. Note however, that if the amount increased by $50%$ every two weeks that would correspond to a larger than doubling every month increase. Compare: $6$ rabbits at start becomes $12$ rabbits after a month: $6$ rabbits at start becomes $9$ rabbits after two weeks becomes $13.5$ rabbits after the second two weeks period. – JMoravitz Jul 08 '16 at 17:06
  • One of the major issues when trying to model growth like this is: do you want the curve to be continuous (might give non-integer answers), or do you want to make the curve discrete (might give periods of time with no growth). If you assume that brand new bouncing baby bunnies are only born at the beginning of each month, then the extra two weeks into the third month will not affect the population. If the population is "large enough" then we might not care so much about making the function discrete. – JMoravitz Jul 08 '16 at 17:11

1 Answers1

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1)Logically, the number of rabbits at any given time should be an integer.

2)You concluded that the function that outputs the number of rabbits as function of time(where the units here are in month) is $$f(t)= 6*2^t $$ This is a false conclusion since number of rabbits cannot be non-integer for any $t$.

3) This problem does not have a unique solution, since the constraints of the problem are not strong enough. For example consider

$$f(t) = \begin{cases} 6*2^t, & t\in\Bbb N\ \\ c, & c\in\Bbb N\ \end{cases}$$

Such a function satisfies the problem. However, one can further impose that $f(t)$ is non-decreasing, so that at any instant in time, the number of rabbits is equal to or greater than the number of rabbits in the past instants, that is $$f(t+\epsilon) \ge f(t)$$

Again, this would not still lead to a unique solution. But one such solution is $$f(t) = \begin{cases} 6*2^t, & t\in\Bbb N\ \\ \dfrac{6*(2^{m+1}+2^{m})}{2}, & m<t<m+1, \quad m\in\Bbb N\ \end{cases}$$

I could come up with other solutions.

So the information given about the problem is not complete.

Omar Nagib
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  • That's where my argument is: I was arguing that the function should be the way I proposed, and the question would then output a non integer answer when plugged into the function, thus proving that the question itself was problematic. That's why I asked if the way I structured my equation was correct or not. – S.Kumar Jul 09 '16 at 00:45