Question

The number of bacteria y (in thousands) in a culture can be approximated by the model y=100(1.99)^t, where t I’d the number of hours the culture is incubated.

Estimate when the number of bacteria will be 1,000,000.

Please help!

Answer 1

It would take about 13.4 hours for the bacteria to reach 1,000,000

Explanation:

We use the exponential growth equation the problem provides, to solve for a total of 1,000,000 bacteria.

Recall that "y"in the equation is the number of individuals at time "t"; 100 is the starting number of the initial population, and t represents the elapsed time (in hours). So "t" i our unknown, and we know all the other values. Since "t" is in the exponent, we proceed to: first isolate the exponential expression, and second, to bring that unknown exponent "t" down by the use of logarithms.

`y=100\,(1.99)^t\\1000000=100\,(1.99)^t\\(1000000)/(100) = (1.99)^t\\10000=(1.99)^t\\log(10000)=t*log(1.99)\\4=t*log(1.99)\\t=(4)/(log(1.99)) \\t=13.3845\, hours`