A culture of bacteria has an initial population of 48000 bacteria and
doubles every 5 hours. Using the formula , where is the
population after hours, is the initial population, is the time in
hours and is the doubling time, what is the population of bacteria in the
culture after 18 hours, to the nearest whole number?

Ask by Gardner Ortiz. in the United States

Mar 21,2025

Question

A culture of bacteria has an initial population of 48000 bacteria and
doubles every 5 hours. Using the formula , where is the
population after hours, is the initial population, is the time in
hours and is the doubling time, what is the population of bacteria in the
culture after 18 hours, to the nearest whole number?

Ask by Gardner Ortiz. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

We start with the formula for exponential growth:

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

where:
- $$ P_0 = 48000 $$ is the initial population,
- $$ t = 18 $$ hours is the elapsed time,
- $$ d = 5 $$ hours is the doubling period.

Substitute the given values into the formula:

$$
P_{18} = 48000 \times 2^{\frac{18}{5}}
$$

Calculate the exponent:

$$
\frac{18}{5} = 3.6
$$

Thus, the expression becomes:

$$
P_{18} = 48000 \times 2^{3.6}
$$

Now evaluate $$ 2^{3.6} $$. We can use the property:

$$
2^{3.6} = e^{3.6 \ln 2}
$$

Given that $$ \ln 2 \approx 0.6931 $$, we have:

$$
3.6 \ln 2 \approx 3.6 \times 0.6931 \approx 2.49516
$$

So,

$$
2^{3.6} \approx e^{2.49516} \approx 12.127
$$

Now, multiply by the initial population:

$$
P_{18} \approx 48000 \times 12.127 \approx 582096
$$

Rounding to the nearest whole number, the population of bacteria after 18 hours is approximately:

$$
\boxed{582096}
$$
After 18 hours, the population of bacteria is approximately 582,096.