Question

part 1 of 2
(a) evaluate p(0) and p(13) and interpret their meaning in the context of this problem.
p(0)=10 means that the population of the bacteria in the culture was 10,000.
p(13)=156.51 means that the 13 hr after the culture was started, 156,510 bacteria were present.
part: 1 / 2
part 2 of 2
(b) predict the population of bacteria 24 hr after the culture was started.
24 hr after the culture was started, bacteria were present.
try one last time

Explanation:

Step1: Assume the population - growth formula

Let's assume the population of bacteria $P(t)$ follows an exponential - growth model $P(t)=P_0e^{kt}$, where $P_0$ is the initial population and $k$ is the growth constant. From $P(0) = 10$, we know that $P_0 = 10$ (in thousands). So $P(t)=10e^{kt}$.

Step2: Find the value of $k$

Since $P(13)=156.51$, we substitute $t = 13$ and $P(13)$ into the formula: $156.51=10e^{13k}$. Then $e^{13k}=\frac{156.51}{10}=15.651$. Taking the natural logarithm of both sides, we get $13k=\ln(15.651)$. So $k=\frac{\ln(15.651)}{13}\approx\frac{2.75}{13}\approx0.2115$.

Step3: Predict the population at $t = 24$

Now we want to find $P(24)$. Substitute $t = 24$, $P_0 = 10$, and $k\approx0.2115$ into the formula $P(t)=P_0e^{kt}$. So $P(24)=10e^{0.2115\times24}$. First, calculate $0.2115\times24 = 5.076$. Then $e^{5.076}\approx159.18$. So $P(24)=10\times159.18 = 1591.8$ (in thousands), which means $P(24)=1591800$.

Answer:

$1591800$