A culture of bacteria has an initial population of 460 bacteria and
doubles every 4 hours. Using the formula , where is the
population after hours, is the initial population, is the time in
nours and is the doubling time, what is the population of bacteria in the
culture after in hours, to the nearest whole number?

Ask by Turnbull Pollard. in the United States

Mar 21,2025

Question

A culture of bacteria has an initial population of 460 bacteria and
doubles every 4 hours. Using the formula , where is the
population after hours, is the initial population, is the time in
nours and is the doubling time, what is the population of bacteria in the
culture after in hours, to the nearest whole number?

Ask by Turnbull Pollard. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

**Step 1: Write the formula with the given information**

The general formula for exponential growth when the population doubles every $$ d $$ hours is

$$
P(t) = P_0 \cdot 2^{\frac{t}{d}},
$$

where  
$$P_0 = 460$$ is the initial bacteria population,  
$$d = 4$$ hours is the doubling time, and  
$$t$$ is the time in hours.

**Step 2: Substitute the given time**

Assuming the question asks for the population after $$ t = 12 $$ hours, substitute the values into the formula:

$$
P(12) = 460 \cdot 2^{\frac{12}{4}}.
$$

**Step 3: Simplify the exponent**

Calculate the exponent:

$$
\frac{12}{4} = 3.
$$

So the expression becomes

$$
P(12) = 460 \cdot 2^3.
$$

**Step 4: Calculate $$ 2^3 $$ and then the population**

We know that

$$
2^3 = 8.
$$

Thus,

$$
P(12) = 460 \cdot 8.
$$

Multiply to obtain:

$$
P(12) = 3680.
$$

**Step 5: Final Answer**

The population of bacteria after 12 hours is $$\boxed{3680}$$ bacteria.
After 12 hours, there are 3,680 bacteria in the culture.