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?

1 Answer
Dec 19, 2017

#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.