A biologist is studying a population of bacteria that triples in size every 4 hours. The p… — SAT

A biologist is studying a population of bacteria that triples in size every 4 hours. The population of the bacteria $t$ hours after the start of the study can be modeled by the function $P(t) = P_0 \cdot 3^{kt}$ , where $P_0$ is the initial population and $k$ is a constant. What is the value of $k$ ?

Answer: 0.25

Answer

The correct answer is 0.25 (or 1/4).

The population triples every 4 hours, meaning that after

t

hours, the population has tripled

\frac{t}{4}

times. Thus, the growth model is

P(t) = P_0 \cdot 3^{\frac{t}{4}}

. Comparing this to the given expression

P(t) = P_0 \cdot 3^{kt}

, we get

kt = \frac{t}{4}

. Dividing both sides by

t

yields

k = \frac{1}{4}

, or

0.25

Step-by-Step Solution

Write the general formula for exponential growth where the population triples every 4 hours.

P(t) = P_0 \cdot 3^{\frac{t}{4}}

If a population triples every 4 hours, it will be multiplied by 3 a total of

\frac{t}{4}

times after

t

hours.

Set the exponent of the general model equal to the exponent of the given model.

kt = \frac{t}{4}

Both models represent the same population growth function, so their exponents must be equal.

Solve for the constant

k

k = \frac{1}{4}

(or

0.25

)

Divide both sides of the equation by

t

Key Concept

Exponential growth functions and representing time intervals in the exponent.

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