Question

a scientist has discovered an organism that produces five offspring exactly one hour after its own birth, and then goes on to live for one week without producing any additional offspring. each replicated organism also replicates at the same rate. at hour one, there is one organism. at hour two, there are five more organisms. how many total organisms are there at hour seven?

a) 19,531
b) 97,655

Explanation:

Step1: Analyze initial - state

At hour one, there is 1 organism.

Step2: Calculate number at hour two

At hour two, the initial 1 organism produces 5 offspring, so the total number of organisms is \(1 + 5=6\).

Step3: Calculate number at hour three

Each of the 6 organisms at hour two produces 5 offspring. So the number of new - born organisms is \(6\times5 = 30\), and the total number of organisms is \(6+30 = 6\times(1 + 5)=6\times6 = 6^{2}=36\).

Step4: Generalize the pattern

We can see that the number of organisms \(N\) at hour \(n\) follows the formula \(N = 6^{n - 1}\) (since at \(n = 1\), \(N = 1=6^{0}\)).

Step5: Calculate number at hour seven

We want to find the number of organisms at hour seven (\(n = 7\)). Using the formula \(N = 6^{7 - 1}=6^{6}\).
\[6^{6}=6\times6\times6\times6\times6\times6=46656\]

Answer:

46656