Question

1. A company has determined that if the price of an item is $ 40, then 150 will be demanded by consumers. When the price is $ 45, then 100 items are demanded by consumers

(a) Find the price-demand equation, assuming that it is linear. ( 2 marks)

(b) Find the revenue function. (1 mark)

(c) Find the number of items sold that will give the maximum revenue. What is the maximum revenue? (4 marks)

(d) What is the price of each item when maximum revenue is achieved? ( 2 mark)

(e) If the company has a fixed cost of $ 3520 and a variable cost of $ 12 per item, find the company's linear cost function. (1 mark)

(f) Find the company's profit function. (2 marks)

(g) Find the number of items sold that will give the maximum profit. What is the maximum profit? (4 marks)

(h) How many items should be sold for the company to break even? (4 marks)

2. Kamal, Ramu and Keong bought 3 types of academic books at a book store. Below is a table showing the number of copies bought for each type.

	Marketing	Management	Finance
Kamal	3	2	5
Ramu	2	3	1
Keong	1	5	4

The total cost of the books for Kamal, Ramu and Keong were RM 267, RM 145 and RM 230 respectively. Using x, y and z to represent the price of each copy of Marketing, Management and Finance book respectively,

(a) obtain a system of linear equations to represent the given information. (3 marks)

(b) write down the system as a matrix equation. (5 marks)

(c) use the Cramer's Rule to solve the system and determine the price of each copy of Marketing, Management and Finance book. (12 marks)

Answer 1

Now a and 0 1 1o. 55 -0.1x (a) Let blad is price when the quantities will be demanded. Let p(x) = atbr - (1) (as it in linear) Þ(150)-40, bl 100) = 45 ( given ) putting these values in (1) = 40= a + 150 b solve for 45 = a + 100b 5 5, ba-1 sol - (2) This in price-demand Equation. (b) Revenue function R(x) þ xx=155-0.12) a SR(X) = 552 -0.142 - (3) (C) RIM= 55-0.2 hon=275 in critical point. R"(M)= -0.220 so x=275 give So Maximum revenue occurs when [275 i teams are sold. Marimum revenue R (275)=558275-0.18(27512 = R(275) = $7562.50 (d) R in maximum when x=275. so price of ca itemo 55-0.1x275 = $ 27.50 =0 6.5 = 275 ves manima. Cach p (e) Cost function C (*)= fixed cost t variable cost ccx) = 3520 +12X (t) Profit + Pla)= Revenue Cost = R(x) -Cla) P19) = 55%-0.17?_3520-12x C Scanned with CamScanne! P(M)= -0.1x? 4 34-3520

215 0.2 (g) for maximum profit: plina 0.2 x +43=0 => X = 43 lo pi (M)= -0.220 do x=2 15 in maximum point. so Profit in maximized when 2 15 items are sold. The maximum profit PL 215)= -0.1(215)2+43x215 -3520 SP(215)= $1102:50 th) whenevere profit is o, the company will break down. lo Plava -0.1X2 +43% -3520 = 0 - 43+ (43)2-4 X-0.1 X-3520 ㅋ x = - 0.2 = 2 = 1 10 or x = 320 when either I do item or 1320 items are sold, the will break. company CScanned with GamScanner

Answer 2

Now a and 0 1 1o. 55 -0.1x (a) Let blad is price when the quantities will be demanded. Let p(x) = atbr - (1) (as it in linear) Þ(150)-40, bl 100) = 45 ( given ) putting these values in (1) = 40= a + 150 b solve for 45 = a + 100b 5 5, ba-1 sol - (2) This in price-demand Equation. (b) Revenue function R(x) þ xx=155-0.12) a SR(X) = 552 -0.142 - (3) (C) RIM= 55-0.2 hon=275 in critical point. R"(M)= -0.220 so x=275 give So Maximum revenue occurs when [275 i teams are sold. Marimum revenue R (275)=558275-0.18(27512 = R(275) = $7562.50 (d) R in maximum when x=275. so price of ca itemo 55-0.1x275 = $ 27.50 =0 6.5 = 275 ves manima. Cach p (e) Cost function C (*)= fixed cost t variable cost ccx) = 3520 +12X (t) Profit + Pla)= Revenue Cost = R(x) -Cla) P19) = 55%-0.17?_3520-12x C Scanned with CamScanne! P(M)= -0.1x? 4 34-3520

215 0.2 (g) for maximum profit: plina 0.2 x +43=0 => X = 43 lo pi (M)= -0.220 do x=2 15 in maximum point. so Profit in maximized when 2 15 items are sold. The maximum profit PL 215)= -0.1(215)2+43x215 -3520 SP(215)= $1102:50 th) whenevere profit is o, the company will break down. lo Plava -0.1X2 +43% -3520 = 0 - 43+ (43)2-4 X-0.1 X-3520 ㅋ x = - 0.2 = 2 = 1 10 or x = 320 when either I do item or 1320 items are sold, the will break. company CScanned with GamScanner