Answer:
(a)[tex]n(t)=9X2^{\frac{2}{3}t}[/tex]
(b)1,482,833
(c)15.2 hours
Step-by-step explanation:
[tex]n(t)=n_02^{t/a}[/tex] where [tex]n_0[/tex] is the initial population, t=time(in hours),
a=interval of growth doubling
(a) [tex]n_0[/tex] =8, a=1.5 hours
After t hours, the number of bacteria in the culture
[tex]n(t)=9X2^{(t/1.5)}\\=9X2^{\frac{2}{3}t}[/tex]
(b)The number of bacteria after 26 hours
[tex]n(26)=9X2^{\frac{2}{3}X26}=9X2^{17.33}=9 X 164759.26\\=1482833[/tex]
After 26 hours, the number of bacteria in the culture will be approximately 1,482,833.
(c)If n(t)=10,000, we want to determine the value of t.
[tex]n(t)=9X2^{\frac{2}{3}t}\\10000=9X2^{\frac{2}{3}t}\\2^{\frac{2}{3}t}=10000/9 \\2^{\frac{2}{3}t}=1111[/tex]
Taking the natural logarithm of both sides
[tex]\frac{2}{3}tX ln(2)=ln 1111\\\frac{2}{3}t=ln1111/ln2\\\frac{2}{3}t=10.12\\t=10.12 X \frac{3}{2}[/tex]=15.2 hours (correct to 1 decimal place)
