Question:

A bacteria population is growing exponentially with a growth rate of 10% per hour. The initial population of bacteria is 500. Write the exponential equation that represents the population at time t. Use this equation to determine the population after 5 hours. Round your answer to the nearest whole number.

Answer:

The exponential equation that represents the population at time t can be written as:

P(t) = P0 * e^(kt)

Where: P(t) is the population at time t, P0 is the initial population, e is the base of the natural logarithm (approximately 2.71828), k is the growth rate (expressed as a decimal), and t is the time.

Given that the growth rate is 10% per hour, the decimal representation of the growth rate, k, can be calculated as follows:

k = 0.10

Substituting the given values into the exponential equation, we have:

P(t) = 500 * e^(0.10t)

To determine the population after 5 hours, we substitute t = 5 into the equation:

P(5) = 500 * e^(0.10 * 5)

Simplifying further:

P(5) = 500 * e^(0.50)

Using a calculator, substitute 0.50 into the equation:

P(5) ≈ 500 * e^(0.50) ≈ 500 * 1.6487212707 ≈ 824.36

Round the answer to the nearest whole number:

P(5) ≈ 824

Therefore, the population after 5 hours is approximately 824.