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1. 1. A communicable disease is spreading through a region and the medical authorities there have carried out research which shows that the number of people infected (N thousand) is a function of the time (t weeks) after the initial occurrence of the disease. They think that the model is

N(t) = t2 + t

(i) Sketch a graph of N against t. (4 marks)

(ii) For what domain of values of t is the model realistic? (2 marks)

(iii) When is the number of people infected a maximum and how many people are infected then?
(2 marks)

(iv) Solve an appropriate equation to find when the number of people infected is equal to 1000? (Answers correct to 3 SF).
(5 marks)

Further research suggests that a more accurate model may be

N(t) = t (t – 5) (t + 4)

(v) Sketch a graph of N against t. (4 marks)

(vi) From your graph, what is the maximum number of people infected? (Answer correct to 3 SF).
(3 marks)

2. Two students, Alan and Betty, work part time washing dishes in a local restaurant. After a particularly busy night each is faced with a mountain of identical dinner plates to wash. They both have their own sink to work at. Water is supplied to each sink at C.

This is too hot for Alan who adds cold water until the temperature of the water in the sink is C. Betty, however, has a pair of rubber gloves and can stand the hotter water. The temperature in the kitchen is C.

Both students are studying an engineering degree course at their university and know that a possible mathematical model for the washing up process is

T(n) = p + qe-0.02n

where T C is the temperature of the water and n is the number of plates washed, while p and q are constants.

(a) Work out the values of p and q

(i) for Alan (ii) for Betty (6 marks)

The students begin washing up and keep going until the water temperature in each sink drops to C.

(b) How many whole plates have been washed.

(i) by Alan (ii) by Betty? (10 marks)

Alan, who has washed fewer plates, resumes the job and continues until Betty’s total has been matched.

(c) What is now the temperature of the water in Alan’s sink?
(4 marks)

3.

V

12.5 -

11.125 -

10.0 -

7.5 -

(not to scale)

0 1 2 3 4 5 t

The graph above shows the variation of voltage, V volts, with time, t seconds, for an electronic device which monitors heat transfer.

It is known that the way V depends upon t is a sinusoidal oscillation about a non-zero mean value.

(a) From the graph work out

(i) the amplitude of the oscillation; (2 marks)

(ii) the mean value of V; (1 mark)

(iii) the period of the oscillation, in seconds; (2 marks)

(iv) the frequency of the oscillation, in cycles per second and in radians per second.
(2 marks)

(b) If V(t) = M + Asin( t + )

(i) write down the values of M, A and ; (3 marks)

(ii) calculate the value of , in radians correct to 3 SF. (6 marks)

(c) Hence find the value of V after 5 minutes, correct to 2 SF. (4 marks)
2. (Original post by RainbowBlood)
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