Question 1

Paper 2Medium7 marks

A laboratory monitors a culture of cells. Its population, yy is measured in thousands of cells, and tt is measured in hours. The rate of change of the population is modelled by

dydt=0.06y(14−y).\frac{\mathrm{d}y}{\mathrm{d}t}=0.06y(14-y).

The table summarizes the slopes in the slope field for this model. Each entry is the value of dydt\frac{\mathrm{d}y}{\mathrm{d}t} at the point given by its row and column. Slopes are in thousands of cells per hour.

yyt=0t=0t=0.5t=0.5t=1t=1
0000
72.942.942.94
111.981.981.98
14000
18−4.32-4.32−4.32-4.32−4.32-4.32
(a)

Show that the slope of the field at (t,y)=(0,9)(t,y)=(0,9) is 2.72.7

[2]
(b)

A culture has y=9y=9 at t=0t=0 Use the gradient at this point to estimate its population at t=0.5t=0.5

[2]
(c)

Use the slope field to decide whether your estimate in part (b) is likely to be greater or less than the modelled population at t=0.5t=0.5 Justify your answer, and state the population towards which the field predicts the culture will tend over a long time.

[3]

More AI 5.3 Differential equations questions