Final answer:
To model the exponential decay of a bacteria culture, utilize the formula y = y_0 · e^{kt}. Given an initial population of 500 and 390 remaining after 6 minutes, solve for the decay constant k using the steps provided to complete the model.
Step-by-step explanation:
To write an exponential decay formula for a population of bacteria that decreases over time, we can use the exponential decay model:
y = y_0 · e^{kt}
Where:
Given that the initial population (y_0) is 500 bacteria and there are 390 bacteria left after 6 minutes, we can calculate the decay constant k using the formula:
390 = 500 · e^{6k}
Divide both sides by 500 to get:
0.78 = e^{6k}
To solve for k, take the natural logarithm of both sides:
ln(0.78) = 6k
Then we can divide by 6 to find k.
Once we have k, we can fill it back into the original decay formula to provide a complete model for the population over time:
y = 500 · e^{kt}
This is the formula relating the number of bacteria to time t, where k is a specific constant that you will calculate based on the initial conditions provided. It allows you to predict the number of bacteria at any given time after the experiment begins.