Question

2 at 7 a.m., a colony of 100 bacteria is placed on a petri dish where the population is growing exponentially and will triple every 6 hours. select all statements that are true about the bacteria population. a. when the bacteria population reaches 900, 12 hours have passed since the colony was placed on the petri dish. b. three hours after the colony is placed on the petri dish, there are 200 bacteria. c. three hours after the colony is placed on the petri dish, there are about 173 bacteria in the colony. d. in the first hour the colony is placed on the petri dish, the population grows by a factor of $3^{\frac{1}{6}}$. e. between 8 a.m. and 9 a.m., the population grows by a factor of $3^{\frac{1}{1}}$.

Explanation:

To solve this, we use the exponential growth formula \( P(t) = P_0 \cdot a^{t/k} \), where \( P_0 = 100 \) (initial population), \( a = 3 \) (triples), \( k = 6 \) (time to triple in hours), and \( t \) is time in hours.

Analyzing Option A:

We need to check if \( P(t) = 900 \) when \( t = 12 \).
Substitute into the formula:
\( P(12) = 100 \cdot 3^{12/6} = 100 \cdot 3^2 = 100 \cdot 9 = 900 \).
This matches, so A is true.

Analyzing Option B:

Check \( P(3) \) (3 hours later).
\( P(3) = 100 \cdot 3^{3/6} = 100 \cdot 3^{0.5} \approx 100 \cdot 1.732 \approx 173 \), not 200.
So B is false.

Analyzing Option C:

From the calculation for B, \( P(3) \approx 173 \).
So C is true.

Analyzing Option D:

The growth factor per hour: The exponent for time \( t \) is \( t/6 \), so the base is \( 3^{1/6} \) (since in 1 hour, \( t = 1 \), so \( 3^{1/6} \)).
Thus, the population grows by a factor of \( 3^{1/6} \) each hour. So D is true.

Analyzing Option E:

Between 8 a.m. and 9 a.m. is 1 hour, so the growth factor is \( 3^{1/6} \), not \( 3^{1/1} \).
So E is false.

The true statements are A, C, D.

Answer:

A. When the bacteria population reaches 900, 12 hours have passed since the colony was placed on the petri dish.
C. Three hours after the colony is placed on the petri dish, there are about 173 bacteria in the colony.
D. In the first hour the colony is placed on the petri dish, the population grows by a factor of \( 3^{\frac{1}{6}} \).