Quantitative Analysis for

Business and Economics

Semester 1, 2014

Assorted Questions

Students are invited to solve these questions and exchange their ideas in the

discussion forum.

1. If a person watches a certain TV daily evening news program on one evening, then the

probability that the person watches that program the next evening is 0.7. However, if the

person does not watch the program one evening, then the probability that the person

watches the program the next evening is 0.2.

(a) If the person watches the program on Monday, what is the probability that the

person watches the program on Wednesday?

(b) If 20% of the population watches the program on Thursday, what percentage can be

expected to watch on Friday?

2. A college dining hall has available two fruit juices for breakfast: orange and grapefruit.

One hundred students drink juice on a daily basis. It is found that a student will not drink

grapefruit juice on two successive days, and if a student drinks orange juice one day, then

the following day the student is equally likely to drink orange juice as to drink grapefruit

juice. If 40 of the regular juice drinkers drink orange juice on Monday, how many can be

expected to drink orange juice on Wednesday?

3. If is an initial state vector for the transition matrix .

Compute the state vector.

4. (from Final exam June 2000):

a) If revenue is given by R = pq where p = price and q = quantity, show using the

product rule that

11 where is point elasticity of demand.

dRp

dq

b) Demand for haircuts in a certain chain of salons is given by the function

200for 11

10q p

p

Where q is the number of haircuts per day in one salon and p is the price of each

haircut. Calculate the elasticity of demand when p = 15.

c) After a 10% GST is introduced, the price of haircuts will rise by the full 10% as there

are no offsetting decreases in wholesale sales tax. Assuming that there is no change

in the demand function, show whether turnover (i.e. revenue including the tax) will

rise or fall when the price increases.

5. A surfboard manufacturer knows that the weekly cost of production, C, and revenue, R,

are functions of output, q, given by:

qqR

qC

40058.0

8001550

2

Evaluate:

a. profit;

b. marginal cost;

c. marginal revenue;

d. average cost when 30 surfboards are produced per week.

6. The Springhill Spa company can sell 3000 litres of mineral water per day if it charges 80c

per litre. For each extra 5c rise in price it sells 100 fewer litres of water per day. What

selling price will maximise the company's income? What is the maximum daily income?

7. A fast food chain has been petitioned by environmentalists to reduce the amount of

polystyrene used in packaging. The company wishes to design a new chicken container

that will minimise the area of polystyrene needed. The box must have a volume of 800

cubic cm. It will be made from a single T shaped piece of material which can be folded

to form a top, bottom, four sides and a tuck-in flap. The flap has the same dimensions as

one of the sides. The top and bottom are equal sized squares. What are the dimensions of

the box that minimise the area of polystyrene used?

8. A monopolist producing a single commodity has a production function given by

q x1 2/

, where q is the quantity produced and x is the amount of the sole input used.

The monopolist can buy any amount of the input at the per unit price of 2, and faces an

industry (inverse) demand function given by , where p is the price per unit of

the output. Assume there are no fixed costs. If all output is sold and the production level

is set to maximise profits, what will the market price be? Also derive an expression for

the marginal revenue product of the input.

9. (From final exam Session 1, 1993)

Suppose that the sales revenue from a particular cultured pearl harvest is (6 + 2.4t) thousand

dollars if the pearls are harvested t years after implantation in the oysters. If the discount rate

is 6% per annum compounded continuously, and the present value of sales revenue is to be

maximised, when should the pearls be harvested and sold (expressed as number of years after

implantation). Do not bother to check the second order condition for a maximum.

10. (Part of Problem 7 from June 1995 exam)

Last year Desert Bloom sold its harvest of 300 tonnes of Nice’N’Salty wheat at a price of

$1800 per tonne. A market study undertaken last year by the firm of accountants Arthur

Sanderson estimated that the price elasticity of annual demand for Nice’N’Salty wheat is

given by

p q 30 3

p

p( )2400 , for p < 2400 where p is the price per tonne.

Assuming the price elasticity expression above is correct, show that the annual demand for

Nice’N’Salty wheat as a function of p is given by

Q

p 1200

2 where Q is the annual production in 3ones.

For each of the following differential equations, find the general solution and the particular

solution corresponding to the given initial condition.

(i) dy

dxx y y 3 0

1

2

2 ,

(ii) dy

dx

e

ey n

x

y , 0 2

(iii)

dy

dx

x

yy

1

2

2

50 7,

11. A country town, affected by the withdrawal of key manufacturing industries, has had a

declining population for a considerable period. The rate of change at any time has been

proportional to the population at that time. At 30th June, 1990 the population was 24,500

and four years later it was 18,000. At the current rate of change how long will it take for

the town’s population to decline to 6,000 people?