Answer
a. $n(t)=25\cdot 2^{t/5}$
b. about 303 bacteria
c. after about $76.4$ hours
Work Step by Step
a. Given: n(0)=25 and n(5)=50 (double after 5 hours)
$50=25\cdot 2^{5/a}$
... solve for a, start by dividing both sides with $25$...
$ 2=2^{5/a}\qquad$ ...apply $\log_{2}$ to both sides
$1=\displaystyle \frac{5}{a}\qquad /\times a$
$a=5$
The model is $n(t)=25\cdot 2^{t/5}$
b. t=$18$,
$ n(t)=25\cdot 2^{18/5}\approx$303.143313302
about 303 bacteria
c. Solve for t when n(t)=1,000,000
$1,000,000=25\cdot 2^{t/5}\qquad /\div 25$
$ 40,000=2^{t/5}\qquad$ ...apply $\log()$ to both sides
$\displaystyle \log 40,000=\frac{t}{5}\log 2\qquad/\times\frac{5}{\log 5}$
$ t=\displaystyle \frac{5\log 40,000}{\log 2}\approx$76.4385618977
after about $76.4$ hours