a study is done on the number of bacteria cells in a petri dish. suppose that the population sizen p(t) after t hours is given by the following exponential function. p(t) =1500(1.07) rase to t%0d%0a

Answer 1

p(t) = 1500(1.07)^t is the formula for the bacteria population after t hours

initial population at time t=0 is 1500 as p(0)= 1500(1.07)^0 = 1500(1) = 1500

doubling time is when 2 =(1.07)^t, solve for t

take logs of both sides, to the base 1.07

log_1.072 = t

t= ln2/ln1.07

t= about 10.245 hours

= about 10 hours, 14 minutes, and 41 seconds

rate of increase per hour = r = .07 = 7%

general formula is p(t) = P(1+r/n)^nt where r=hourly rate of increase, t=hours, n=compounding periods per year

but for bacteria

the usual formula involves continual compounding

P(t) = Pe^rt where e is an irrational constant = about 2.718281828

P(0) = 1500e^rt

double time t = ln2/ln1.07

2 = about e^10.245r

ln2 = 10.245r

r = (ln2)/10.245 = about .06766 = 6.8% hourly rate of increase

P(t) =1500e^.06766t is equivalent to P(t)=1500(1.07)^t