Question

I need some help with this Calculus problem.

A bacteria population starts with 500 bacteria and grows at a rate of r(t) = 548e^6.5t bacteria per hour.

A. Determine the function P(t) which gives the population at time t.

C. How long does it take the initial population to triple in size?

Answers

Damon
dP/dt = 548 e^(6.5 t) ????
if I understand you that means
P = (548/6.5) e^(6.5 t) + constant
but we know when t = 0, P = 500
so
500 = 84.3 + c
c = 415.7
so
P = 84.3 e^(6.5 t) + 415.7

when is it 1500?

1500 - 415.7 = 84.3 e^(6.5 t)
12.86 = e^6.5 t
ln 12.86 = 6.5 t
Reiny
Since r(t) is the rate,
r(t) = 548e^6.5t is the derivative of P(t) , then
P(t) = 548/6.5 e^6.5t + c , where c is a constant
when t = 0, P(0) = 500
500 = 548/6.5 (e^0) + c , but e^0 = 1 , so ....
c = appr 415.69

P(t) = (548/6.5) e^6.5t + 415.69

Then when is P(t) = 1500 ?

1500 = (548/6.5) e^6.5t + 415.69
e^6.5t = 12.86131...
6.5t = ln 12.86131...
t = .393 hrs or appr 23.6 minutes
James
Awesome, thank you! There is one more part to this question I need help with. I tried figuring it out but the number of bacteria I calculated didn't make any sense. It says to determine the number of bacteria after one hour.
Damon
P = 84.3 e^(6.5 t) + 415.7
if t = 1
P(1) = 84.3 e^6.5 + 415.7
= 56,487

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