Question

A culture of bacteria has an initial population of 12000 bacteria and doubles every 6 hours. Using the formula P_t = P_0\cdot 2^{\frac{t}{d}}P t ​ =P 0 ​ ⋅2 d t ​ , where P_tP t ​ is the population after t hours, P_0P 0 ​ is the initial population, t is the time in hours and d is the doubling time, what is the population of bacteria in the culture after 17 hours, to the nearest whole number?

Answer 1

Answer:

85,524 bacteria

Step-by-step explanation:

Given the population of bacteria modelled by the equation where;

$P_t = P_0\cdot 2^{\frac{t}{d}$

P0 is the initial population

t is the time in hours

d is the doubling time

If the culture of bacteria has an initial population of 12000 bacteria, this means that;

at t = 0, P0 = 12000

substitute into the modelled function

$P_t = P_0\cdot 2^{\frac{t}{d}\\\\P_t = 12000\cdot 2^{\frac{0}{d}\\\\\\P_t =12000\cdot 2^{0}\\\\P_t = 12000$

If the population doubles every 6 hours

at t = 6, Pt = 24000

$P_t = P_0\cdot 2^{\frac{t}{d}\\\\\\24000 = 12000\cdot 2^{\frac{6}{d}\\\\\\\\24000/12000 = 2^{\frac{6}{d}$

$2 = 2^{\frac{6}{d} }\\1 = 6/d\\d = 6$

Next is to get the population of bacteria in the culture after 17 hours

at t = 17, Pt = ?

$P_t = P_0\cdot 2^{\frac{t}{d}\\\\P_t = 12000\cdot 2^{\frac{17}{6}\\\\\\\\P_t = 12000\cdot 2^{ 2.833 }\\P_t = 12000\cdot 2^{ 2.833 }\\P_t = 12000 * 7.127\\P_t = 85,524.3$

Hence the population of bacteria in the culture after 17 hours, to the nearest whole number is 85,524 bacteria