Question

I'm studying a new bacteria that doubles in numbers every 25 minutes. If i start with 50 bacteria, how long until i have 5 million of them

Answer 1

415.63 minutes

Explanation:

Growth can be represented by the equation
`A=A_0e^(rt)`. We can find the rate at which it grows by using t=25 minutes and
`(A_(0))/(A) =2` or double the amount at that time. The first step we always take is to divide
`A_0` by A.

`[tex](A_(0))/(A)=e^(rt)\\2=e^(r(25))`

`2=e^((25)r)`

To solve for r, we will take the natural log of both sides and use log rules to isolate r.

`ln 2=ln e^((25)r)\\ln 2=25r (ln e)\\(ln2)/(25) =r`

We know
`lne=1` so we were able to cancel it out and divide both sides by 25.

We solve with a calculator
`(ln2)/(25) =r\\0.0277=r`

We change 0.0277 into a percent by multiplying by 100 to get 2.77% as the rate.

The equation is
`A=A_0e^(0.0277t)` .

We repeat the step above substituting A=5,000,000,
`A_0`=50, and r=0.02777. Then solve for t.

`5000000=50e^(0.0277t)\\(5000000)/(50) =e^(0.0277t)\\100000=e^(0.0277t)\\ln100000=lne^(0.0277t)\\ln100000=0.02777t(lne)\\(ln100000)/(0.0277) =t`

t=415.63 minutes