Cambridge International Mathematics

670 Investigation and modelling questions (Chapter 34)

4 The surge model has form y=

at

2 bt

whereaandbare constants andtis the time,t> 0 :

This model has extensive use in the study of medical doses where there is an initial rapid increase to a

maximum and then a slow decay to zero.

a Use a graphics calculator to graph the model (on the same set of axes) for:

i a=10, b=2 ii a=15, b=3

Time (thours) 0 2 4 6 12

Effect (Eunits) 0 25 25 r s

b The effect of a pain killing injectionthours after

it has been given is shown in the following table:

The effectEfollows a surge model of the form

E=

at

2 bt

.

i By usingtwoof the points of this table, find the values ofaandb.

ii Hence, find the values ofrandsin the table.

iii Use your calculator to find the maximum effect of the injection and when it occurs.

iv It is known that surgical operations can only take place when the effectiveness is more than

15 units. Between what two times can an operation take place?

5 The logistic model has form y=

a£ 2 bt

2 bt+c

wheretis the time, t> 0. The logistic model is useful

in describing limited growth problems, i.e., when theyvariable cannot grow beyond a particular value

for some reason.

a Use technology to help graph the logistic model for a=3, b=^12 and c=2.

(Use the window ¡ 16 x 640 , ¡ 16 y 65 .)

b What feature of the graph indicates a limiting value?

c What is the limiting value?

d Bacteria is present in a carton of milk and afterthours the bacteria (Bunits) was recorded as

follows:

t 0 1 2

B(t) 10 12 : 70 15 : 03 It is known that c=1.

i Use the first two sets of data to findaandb, and hence determine the logistic model.

ii Use the model found inito check the third data set.

iii What is the limiting quantity of bacteria for this model?

iv In the general model y=

a£ 2 bt

2 bt+c

, explain why the limiting quantity has valuea.

6 June 1991, Paper 4

A farmer keepsxgoats andycows. Each goat costs$2a day to feed and each cow costs$4a day to

feed. The farmer can only afford to spend$32a day on animal food.

a Show that x+2y 616 :

b The farmer has room for no more than 12 animals. He wants to keep at least 6 goats and at least

3 cows. Write down three more inequalities.

c Using a scale of 1 cm to represent 1 unit on each axis, represent the four inequalities on a graph.

d One possible combination which satisfies all the inequalities is 6 goats and 4 cows. Write down

all the other possible combinations.

e If he makes a profit of$50on each goat and$80on each cow, which combination will give him

the greatest profit? Calculate the profit in this case.

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Y:\HAESE\IGCSE01\IG01_34\670IGCSE01_34.CDR Thursday, 6 November 2008 3:55:37 PM PETER