Question

the small intestine bacteriaa, while inhabiting areas optimal for growth have a doubling time of roughly 10 hours. A normal small intestine starting population would be approximately 10,000 bacteria per ml of fluid

a) write an equation to model this exponential growth, with b(x) representing the number of bacteria per ml and x representing the time in hours

b) how long will it take for there to be 100,000 bacteria per ml (round answer to two decimal places in necessary)

c) determine the average rate of change between 20 hours and 30 hours

d) estimate the instantaneous rate of change at 24 hours

Answer 1

b(x) = 10000(2)^(x/10)

b) when b(x) = 100000
10000(2)^(x/10) = 100000
2^(x/10) = 10
take log of both sides and use log rules
(x/10)log2 = log 10
x/2 = 1/log2
x = 2/log2 = ..... , don't have my calculator handy

c) find b(30) and b(20)
avg rate of change = (b(30) - b(20))/(30-20) = ....

d) do something like
rate = (b(24.01) - b(24))/(24.01-24)
use your calculator

Answer 2

for b), i did it like

100000 = 10000(2)^(x/10)
10 = 2^(x/10)
log 10 = x/10 log 2
log 10/log 2 = x/10
3.3219 = x/10
33.22 = x

would this still be correct?

Answer 3

for b) yes, you are correct.
I have a typo, I did x/2 instead of x/10
Can't explain why I did that, just a typo I guess.