Question

In a laboratory experiment, the population of bacteria in a petri dish started off at 7500 and is growing exponentially at 14% per day. Write a function to represent the population of bacteria after t days, where the hourly rate of change can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of change per hour, to the nearest hundredth of a percent.

Answer 1

Explanation:

The function to represent the population of bacteria after t days can be written as:

P(t) = P0 * (1 + r)^t

where P0 is the initial population (7500), r is the daily growth rate (14% = 0.14), and t is the number of days.

Substituting the values, we get:

P(t) = 7500 * (1 + 0.14)^t

Simplifying, we get:

P(t) = 7500 * 1.14^t

To find the percentage rate of change per hour, we need to convert the daily growth rate to an hourly growth rate. Since there are 24 hours in a day, the hourly growth rate, denoted by h, is given by:

h = (1 + r)^(1/24) - 1

Substituting the value of r, we get:

h = (1 + 0.14)^(1/24) - 1

= 0.005521

So, the percentage rate of change per hour is:

h * 100% = 0.5521%

Rounding to the nearest hundredth of a percent, we get:

h * 100% ≈ 0.55%