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A related rates math problem?
A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 290.
(a) Find an expression for the number of bacteria after t hours.
(b) Find the number of bacteria after 2 hours.
(c) Find the rate of growth after 2 hours.
(d) When will the population reach 10,000?
Can someone do these and show me the steps? I've tried them already, but all of my answers are wrong, I must have the steps wrong. Thanks in advance!
Also, for part B, the growth rate should be in cells per hour. Otherwise, David's answer was very very helpful :)
1 Answer
- 1 decade agoFavorite Answer
This follows the standard exponential growth formula:
y=K(1+r)^t, where y is the amount at time "t", K is the initial amount at time (0) and r is the growth factor.
For your data, we have 290=100(1+r)^1, solving for r, we find that the growth factor is 1.9.
for y(2), y=100(1+1.9)^2 = 841
at two hours, the growth rate=(1+1.9)^2=8.41
to determine when the population reaches 10000:
y=10000=100(1+1.9)^t >> 100=(2.9)^t
take the natural log of both side to solve for t:
ln(100)=t*ln(2.9) and solve for t >> t=ln(100)/ln(2.9)=4.605/1.065=4.325 hours
