Question

If a bacteria population starts with 80 bacteria and doubles every three hours, then the number of bacteria after t hours is n = f(t) = 80 · 2t/3.
(a) Find the inverse of this function. Explain its meaning.
(b) When will the population reach 10,000? (Round your answer to one decimal place.)

Answer 1

Answer:

$f^{-1}(x)=3\frac{ln(\frac{t}{80})}{ln(2)}$

It will take 20.9 hours to reach 10000

Step-by-step explanation:

$f(t)=80(2)^{\frac{t}{3}}$

To find inverse of the function , replace f(x) with y

$y=80(2)^{\frac{t}{3}}$

swap the variables

$t=80(2)^{\frac{y}{3}}$

solve the equation for y

$t=80(2)^{\frac{y}{3}}\\\\\frac{t}{80} =(2)^{\frac{y}{3}}$

take ln on both sides

$ln(\frac{t}{80} )= \frac{y}{3} ln(2)\\\\\frac{ln(\frac{t}{80} }{ln(2)} =\frac{y}{3}\\3\frac{ln(\frac{t}{80} }{ln(2)} =y$

the inverse function is

$f^{-1}(x)=3\frac{ln(\frac{t}{80})}{ln(2)}$

for part b plug in 10000 for t in f^-1(x)

$f^{-1}(10000)=3\frac{ln(\frac{t}{80} }{ln(2)} \\f^{-1}(10000)=3\frac{ln(\frac{10000}{80} }{ln(2)} \\=20.89$

It will take 20.9 hours to reach 10000