Question

IN EXCEL: 5.2.1 At time t = 0, a yeast culture weighs 0.5 g. Two hours later, it weighs 2 g. The maximum weight of the culture is 8 g.

1. Create a spreadsheet to model the population using a logistic equation. Use a scroll bar to vary the value of k.

2. Use the scroll bar to find a value of k so that the condition y(2) = 2 is satisfied.

3. At what time is the weight increasing most rapidly? Support your answer numerically.

Answer 1

It is given that at t = 0, P(0) = 0.5. This means that Also, the maximum weight of the culture = 8g. In other words k = 8.

At t = 2, P(2) = 2. Using this we can find the value of r.

(2) Now to create a spreadsheet model, we'll define the constants k, r and P_0 and add 3 columns for time, population and rate of change of population (dP/dt).

(a) We can define the step size for the time variable. In this case, it is 0.1 hour or 6 minutes.

(b) We can use cell referencing for defining the logistic growth formula.

(c) Now, we'll vary the value of constant k using a scroll bar.

(d) We'll use the developer tab to insert the scroll bar. Its function is to the change the value of the cell to which it is assigned.

(e) Now, right-clicking on the scroll bar will open a tab, select format control. And in the 'Cell link' row, we'll assign the cell in which we are storing the value of k. In this case A3.

(f) We can also change the max. and min. value on the scroll bar. In this case, k lies in [1, 20].

(3) Careful observation will reveal that dP/dt is maximum when P(t) = k/2.

For k =20

For k = 6