Question:

A bacteria culture initially contains 200 bacteria. The number of bacteria in the culture doubles every 3 hours.

(a) Write an exponential growth function to model the population after time t in hours.

(b) Determine the population of bacteria in the culture after 12 hours.

Answer:

(a) To write the exponential growth function, we need to express the population as a function of time. Let P(t) represent the population after time t in hours. Since the population doubles every 3 hours, we can write the equation:

P(t) = 200 * 2^(t/3)

The base of the exponential function is 2, as the population doubles, and the exponent is t/3 since the population doubles every 3 hours.

(b) To determine the population after 12 hours, we substitute t = 12 into the exponential growth function:

P(12) = 200 * 2^(12/3)

Simplifying this equation, we have:

P(12) = 200 * 2^4

P(12) = 200 * 16

P(12) = 3200

Therefore, the population of bacteria in the culture after 12 hours is 3200.

Explanation:

(a) The exponential growth function is derived from the fact that the population doubles every 3 hours. We start with an initial population of 200 bacteria, and for every 3 hours that pass, the population doubles. This pattern is reflected in the equation P(t) = 200 * 2^(t/3), where the base 2 represents the doubling effect.

(b) To find the population after 12 hours, we substitute t = 12 into the growth function. Evaluating this equation, we simplify the expression to find that the population after 12 hours is 3200 bacteria. This result can be found by calculating P(12) = 200 * 2^4 = 3200.