Answer:
Step-by-step explanation:
We know that:
If we go 8 months before, the population will divide by 2.
Conversion:
Hence, before 2 years, the population was below 10 mice.
aka 24 months
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Explanation:
The population follows this function
y = 72*(2)^x
where x is the number of eight-month periods and y is the population.
So if for instance we had x = 2, then it represents two periods of 8 months each, giving 2*8 = 16 months total.
Normally with timescale problems like this, negative time values do not make sense. However, having x be negative will allow us to look back in time to figure out when y < 10.
Let's see what x is when y = 10.
y = 72*(2)^x
10 = 72*(2)^x
10/72 = 2^x
0.1389 = 2^x approximately
log(0.1389) = log(2^x)
log(0.1389) = x*log(2)
x = log(0.1389)/log(2)
x = -2.8479 also approximate
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If x = -2, then,
y = 72*(2)^x
y = 72*(2)^(-2)
y = 18
and if x = -3, then,
y = 72*(2)^x
y = 72*(2)^(-3)
y = 9
This means that 3 eight-month periods ago, the population was less than 10 mice.
So this is 3*8 = 24 months = 2 years.
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Checking the answer:
If we were to start at 9 mice and double the population, then it goes to 18 mice. Then that doubles to 36, and so on.
Here's a table showing this process
[tex]\begin{array}{|c|c|} \cline{1-2} \text{Time In Months} & \text{Population}\\ \cline{1-2} \text{0} & \text{9}\\ \cline{1-2} \text{8} & \text{18}\\ \cline{1-2} \text{16} & \text{36}\\ \cline{1-2} \text{24} & \text{72}\\ \cline{1-2} \end{array} [/tex]
The table confirms the answer of 24 months = 2 years.
The process of using logs is almost like reversing this process shown in the table.
