A population of bacteria in a petri dish 𝑡 hours after the culture has started is given by 𝑃 is 2,400 multiplied by 𝑒 raised to the power 0.084𝑡. Amelia says this means that the growth rate is 8.4 percent per hour. Her friend Elizabeth, however, says that the hourly growth rate is 8.76 percent. Who is right?
We’re given an exponential function describing the growth of a population of bacteria. That’s 𝑃 is equal to 2,400 multiplied by 𝑒 raised to the power 0.084𝑡, where 𝑡 is time in hours. We want to find the growth rate per hour as a percentage to determine who is correct, Amelia or Elizabeth.
Now we know that for positive values of 𝑘 and for the initial value 𝑎, the function 𝑦 of 𝑡 is equal to 𝑎 multiplied by 𝑒 raised to the power 𝑘𝑡 represents exponential growth. We can also express this as 𝑦 of 𝑡 is equal to 𝑎 multiplied by 𝑏 raised to the power 𝑡, where 𝑏 is greater than one. The correspondence between these two expressions of exponential growth is that 𝑘 is equal to ln 𝑏. That’s ln 𝑏, the natural algorithm of 𝑏.
Taking exponentials on both sides, this gives us 𝑒 raised to the power 𝑘 is 𝑒 raised to the power ln 𝑏. And since logarithms and exponentials are inverse functions, we have the correspondence that 𝑒 raised to the power 𝑘 is equal to 𝑏. Now comparing with the growth of the population of bacteria, we have 𝑘 is equal to 0.084. And this means that 𝑏 in our exponential growth equation is equal to 𝑒 raised to the power 0.084.
Now in terms of a constant rate of change, that’s uppercase 𝑅, we have 𝑏 is equal to one plus 𝑅, where 𝑅 is greater than zero implies growth. And now making 𝑅 the subject by subtracting one from both sides, we have 𝑏 minus one is equal to 𝑅. Now since 𝑅 is the proportionate rate of change, as a percentage we have lowercase 𝑟 is 100 multiplied by uppercase 𝑅. And that’s equal to 100 multiplied by 𝑏 minus one, since we found that 𝑏 minus one is uppercase 𝑅. And for our population of bacteria, 100 multiplied by 𝑏 minus one is 100 multiplied by 𝑒 raised to the power 0.084 minus one.
Typing 𝑒 to the power 0.084 into our calculators gives us 1.08762 and so on. And evaluating our parentheses, we have 𝑟 is equal to 100 multiplied by 0.08762 and so on. Evaluating this to two decimal places gives us the growth rate then of 8.76 percent per hour. And hence, we can conclude that Elizabeth is right. If a population of bacteria in a petri dish 𝑡 hours after the culture has started is given by 𝑃 is equal to 2,400 multiplied by 𝑒 raised to the power 0.084𝑡, then the hourly growth rate is 8.76 percent.