Question

the population of bacteria in a culture can be modeled by ( p(t)=-0.01 t^{3}+12.96 t + 10 ), where ( t ) is the time in hours after the culture was started. the population in thousands.

population of bacteria versus time

part: ( 0 / 2 )

part 1 of 2

(a) evaluate ( p(0) ) and ( p(12) ) and interpret their meaning in the context of this problem.

( p(0)=10 ) means that the initial population of the bacteria in the culture was

( p(12)=148.24 ) means that the 12 hr after the culture was started, 148240 bacteria were present.

Explanation:

Step1: Evaluate \(P(0)\)

Substitute \(t = 0\) into \(P(t)=-0.01t^{3}+12.96t + 10\).

$$
LATEXBLOCK0
$$

Since \(t = 0\) represents the start - time of the culture, \(P(0) = 10\) means the initial population of bacteria (in thousands) is \(10\), so the number of bacteria is \(10\times1000=10000\).

Step2: Evaluate \(P(12)\)

Substitute \(t = 12\) into \(P(t)=-0.01t^{3}+12.96t + 10\).

$$
LATEXBLOCK1
$$

Since \(P(t)\) is the population in thousands, the number of bacteria when \(t = 12\) (12 hours after the culture was started) is \(148.24\times1000 = 148240\).

Answer:

\(P(0)=10\) means that the initial population of the bacteria in the culture was \(10000\). \(P(12)=148.24\) means that \(12\) hours after the culture was started, \(148240\) bacteria were present.