Unlimited growth: Exponential growth
Formulas for exponential growth
For an exponential growth model, when you know the value of quantity at two times, then you can calculate the constant and growth factor in the corresponding formula .
Suppose and are the values of a variable at time and at a later time . The growth factor per time interval equals:
An example clarifies this calculation:
Suppose that you took a chicken leg from the refrigerator at one o'clock in the afternoon and put it on a plate to reach room temperature.
Suppose room temperature was reached after two hours and the number of salmonella bacteria on the meat grows exponentially after this moment.
At six o'clock in the evening there are a hundred thousand bacteria and at nine o'clock in the evening, this number increased to 1.6 million.
Give the formula for the number of bacteria , in which is the time in hours, with at three o'clock in the afternoon.
Round the growth factor to four decimal numbers.
Explanation
This corresponds to a doubling time of 3/4 hours = 45 minutes.
, and leads to the equation .
So .
We have found the following formula:
Check: from the data you can see that the number of bacteria is multiplied by sixteen every three hours. Three hours before the clock strikes six, accidentally time , the number of bacteria is 16 times less than the given number at six o'clock. So .
In general, two data points are needed to find the parameters and in the general formula . These may be two measurements, but also, for example a single measurement value, and the doubling time of the growth process. It is even conceivable that two growth rates are known and are to be used to calculate the parameter values.