Question

The number of mold spores in a petri dish increases by a factor of 10 every week. If there are initially 40 spores in the dish, how long will it take for there to be 2000 spores?

Answer 1

`0.8495` weeks

5 days, 22 hours ( to the nearest hour )

Explanation:

We need to find an equation of the form:

`A(t)=A_0e^(kt)`

Where `A_0` is the initial amount, `k` is the growth/decay factor, `A(t)` is the amount after time `t` and `t` is the time. For this example we will take `t` to be in weeks.

From the given information we know the initial amount is 40.

If they are increasing by a factor of 10 every week, then we would expect the amount after 1 week to be 400. So, using our equation:

`400=40e^(k)` ( t = 1 for 1 week)

We need to solve this to find the growth/decay factor `k`.

Divide both sides by 40:

`100=e^k`

Taking natural logs of both sides:

`ln(100)=kln(e)` **( ln(e)=1, the logarithm of the base is always 1)
**
`k=ln(100)`

Now we know `k` we can solve the problem for `t`:

Final amount is `2000`, So:

`2000=40e^(ln(100)t)`

Using the fact that `e^ln(a)=a`

`2000=40(100)^t`

Divide by 40:

`50=(100)^t`

Taking logs of both sides:

`ln(50)/ln(100)=t=>t=0.8495` weeks (4 .d.p)

or 5 days, 22 hours ( to the nearest hour )

CHECK:

`A(t)=40e^(ln(100)t)`

`A(t)=40e^(ln(100)*0.8495)`

`A(t)=40*(100)^0.8495=2000.13814`

We wouldn't expect this to be exact, because we rounded to 4 dp.