Question 1159189
The count in a bacteria culture was 600 after 10 minutes and 2000 after 35 minutes.
 Assuming the count grows exponentially,
Using the form that a*b^x = y
x=10, y=600
{{{a*b^10 = 600}}}
{{{a = 600/b^10}}}
x=35, y=2000
{{{a*b^35 = 2000}}}
replace a with {{{600/b^10}}}
{{{(600/b^10)*b^35 = 2000}}}
cancel b^10
{{{600*b^25 = 2000}}}
{{{b^25 = 2000/600}}}
{{{b^25 = 20/6}}}
{{{25*ln(b) = ln(3.333)}}}
ln(b) = ln(3.333)/25}}}
ln(b) = .048159
b = 1.049337
Find a
a = {{{600/1.048337^10}}}
a = {{{600/1.6186}}}
a = 370.7
The equation
{{{y = 370.7*1.049337^x)}}}
:
What was the initial size of the culture?
x=0, therefore 370.7 is the initial amt
:
Find the doubling period.
{{{1.049337^x = 2}}}
x*ln(1.049337) = ln(2)
{{{x = ln(2)/ln(1.049337)}}}
x = 14.4 min
:
Find the population after 120 minutes.
{{{y = 370.7*1.049337^120)}}}
y = 370.7*323.443
y = 119,900.5
:
When will the population reach 10000
{{{370.7*1.049337^x = 10000}}}
{{{1.049337^x = 10000/370.7}}}
{{{1.049337^x = 26.976}}}
x*ln(1.049337) = ln(26.976)
{{{x = ln(26.976)/ln(1.049337)}}}
x = 68.42 min
:
looks like this
{{{ graph( 300, 200, -50, 150, -20000, 100000, 370.7*1.049337^x) }}}