Questions: In a laboratory experiment, the population of bacteria in a petri dish started off at 850 and is growing exponentially at 8% 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.

Solution Steps

Let the initial population of bacteria be \( P_0 = 850 \). The daily growth rate is given as \( r = 0.08 \).

To find the hourly growth rate, divide the daily growth rate by the number of hours in a day: \[ \text{hourly growth rate} = \frac{r}{24} = \frac{0.08}{24} = \frac{1}{720} \approx 0.0033333 \]

The hourly growth factor can be calculated as: \[ \text{hourly growth factor} = 1 + \text{hourly growth rate} = 1 + 0.0033333 \approx 1.0033333 \]

The population after \( t \) days can be expressed using the exponential growth formula: \[ P(t) = P_0 \cdot (\text{hourly growth factor})^{24t} \]

Substituting \( t = 5 \) into the formula: \[ P(5) = 850 \cdot (1.0033333)^{120} \approx 1267.2078 \]

The percentage rate of change per hour is given by: \[ \text{percentage rate per hour} = (\text{hourly growth factor} - 1) \cdot 100 = (1.0033333 - 1) \cdot 100 \approx 0.33\% \]

Final Answer