QuestionQuestion 2
A bacteria culture initially contains 1500 bacteria and doubles every half hour.
Find the size of the baterial population after 60 minutes. $\square$
Find the size of the baterial population after 9 hours. $\square$ Question Help: Video Message instructfory Submit Question

Studdy Solution

STEP 1

1. The initial bacteria population is 1500.
2. The bacteria population doubles every half hour.
3. We need to find the population after 60 minutes (1 hour).
4. We need to find the population after 9 hours.

STEP 2

1. Determine the number of doubling periods.
2. Calculate the population after 60 minutes.
3. Calculate the population after 9 hours.

STEP 3

Determine the number of doubling periods.
For 60 minutes: Since the population doubles every half hour, in 60 minutes (1 hour), there are:
$\frac{60}{30} = 2 \text{ doubling periods}$
For 9 hours: Convert hours to minutes first, then determine the number of doubling periods:
$9 \text{ hours} = 9 \times 60 = 540 \text{ minutes}$
$\frac{540}{30} = 18 \text{ doubling periods}$

STEP 4

Calculate the population after 60 minutes.
The formula for population growth with doubling is:
$P = P_0 \times 2^n$
where $P_0$ is the initial population and $n$ is the number of doubling periods.
For 60 minutes:
$P = 1500 \times 2^2$ $P = 1500 \times 4$ $P = 6000$
The size of the bacterial population after 60 minutes is:
$\boxed{6000}$

STEP 5

Calculate the population after 9 hours.
Using the same formula:
For 9 hours:
$P = 1500 \times 2^{18}$
Calculate $2^{18}$ :
$2^{18} = 262144$
Now calculate the population:
$P = 1500 \times 262144$ $P = 393216000$
The size of the bacterial population after 9 hours is:
$\boxed{393216000}$

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