Answer:
Future amount = 48(1 + 0.316074)^12
Explanation:
You want the exponential equation that can be used to find the number of bacteria in a colony after 12 days if it starts with 48 bacteria and triples in 4 days.
Exponential function
The exponential function can be written in the form ...
future amount = a·b^(t/p)
where 'a' is the initial amount, 'b' is the growth factor in period p.
The problem statement tells you the initial amount is 48, and the factor by which the amount grows is a factor of 3 in 4 days. This lets you write the function as ...
future amount = 48·3^(t/4) . . . . . where t is in days.
Rearrangement
This equation can be rearranged so the only exponent is t:
future amount = 48·(3^(1/4))^t = 48·1.316074^t
future amount = 48·(1 +0.316074)^t
Then the amount in 12 days is ...
future amount = 48·(1 +0.316074)^12
The numbers that go in the boxes are 0.316074 and 12.
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Additional comment
Working from our original equation, we would compute the number after 12 days to be ...
future value = 48·(3)^(12/4)
Putting the growth factor in "1 +" form, this could be ...
future value = 48·(1 +2)^3
and your box numbers would be 2 and 3. We doubt your answer checker will accept this version of the equation.
Either way, the number of bacteria after 12 days is 1296.
If you use 3^(1/4) -1 as the growth rate in the first box, you need to report it to at least 5 significant digits in order for the future value to come out right. We have shown the value with 6 significant figures.
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