2 = -30 or 20 Hence an output of 2000 will maximize profit.

Transcription

1 QUANTITATIVE TOOLS IN BUSINESS NOV. SOLUTION (a) (i) Marginal Cost MC = Q Q + Total Cost TC = ʃ Mc = ʃ (Q Q + ) = Q Q + Q + C Where C is the constant of integration (ie C is fixed cost in GHS ) When Q =, TC = ie = () () + () + C = Hence the fixed cost is GHS,, (ii) Total cost : TC = Q Q + + Unit Price : P = Q Total Revenue : TR = P Q = Q Q TI = TR TC = Q Q Q + Q dπ = Q Q + For maximum profit, dπ = ie -Q Q + = or Q + Q = For ax + bx + C = ; x = -b = \ ac a where x = Q, a =, b =, c = - :. Q = - + \ - () (-) () = + = - or Hence an output of will maximize profit. NB Accept Marginal cost = Marginal revenue approach. Page of

2 QUANTITATIVE TOOLS IN BUSINESS NOV. (iii) The point elasticity of demand is: PED = P Q dp where Q = P = () = dp = - :. PED = (-) =. (b) Total tax: Ta = Q Profit after tax: Πa = Π Ta = -Q Q + Q dπa = -Q + At maximum Πa, dπa = ie -Q - Q + = or Q + Q - = :. Q = - + \ () (-) () =. or -. Choose Q =. since Q Item Before Tax After Tax Remarks Output Price (P) Profit (Π) 9 Decrease Increase Decrease Page of

3 QUANTITATIVE TOOLS IN BUSINESS NOV. SOLUTION (a) Mutually Exclusive events: Two events are mutually exclusive if the occurrence of either event excluded the possibility of the occurrence of the other event, that is, either one event or the other event but not both can occur. Independent Events: Two events are independent if the occurrence or own occurrence f one event has no influence on the occurrence or non occurrence of the other event. (b) Probabilities at pick no o /9 no /9 /9 o no /9 o / / no o / no / o / no / o / no / x x 9 x x 9 x x 9 x x 9 x x 9 x x 9 x x 9 x x 9 (EEE) (EEE) (EĒE) (EĒĒ) (ĒEE) (ĒEĒ) (ĒĒE) (ĒĒĒ) Page of

4 QUANTITATIVE TOOLS IN BUSINESS NOV. (i) P (EEE) = x x = 9 (ii) P (one in only) = x x + x x + x x = /9 (iii) P (two in only) = x x + x x + x x = /9 (iv) P (non in ) = x x 9 = / (c) Assume we have the same number of erroneous and good invoice Let E be event erroneous increase G be event Good invoices C be correctly picked invoices (C) = (C E) + (G C) = (C/E). (E) + (C/G). (G) (C/E) =. (C/G) =. (C) =. (G) =. (C) =. x. +. x. =. (i) (All three are erroneous) = (.) (.) - = (.) (ii) (Only one is erroneous) = (.) (.) - =. Page of

5 QUANTITATIVE TOOLS IN BUSINESS NOV. (iii) (Only two are erroneous) = (.) (.) - =.9 (iv) (None is erroneous) = (.) (.) - =. SOLUTION (a) Year Month Data value Cumulative Total Total for year ending at stated month 9 January February March April May June July August September October November December Graph for Z- Chart Page of

6 QUANTITATIVE TOOLS IN BUSINESS NOV. Year Month Production levels( ) (Y) January February March April 9 May June July 9 August September October November 9 December 9 January February March April May June July August 9 September October November December MT MY Centred MA (T) Incremental trend =.. Forecast for : =. Month Trend (T) S Forecast (Ŷ) January. + x. =. -.. February. + 9 x. = Page of

7 QUANTITATIVE TOOLS IN BUSINESS NOV. SOLUTION (a) Let the quantity of goods sold be Q, selling price of the goods be P, and the cost of the goods be C Then Q =, P =..., C =... Total revenue, TR = P T Q = (...) =, Total cost, TC = CTQ = (...) =, Profit = Total Revenue Total Cost =, -, =,9 Profit for the week is = GHS,9 (b) P + P = P + P = Let A = det A = ( x ) - () () = = adj A = - - :. A- = adj A det A = - - Page of

8 QUANTITATIVE TOOLS IN BUSINESS NOV. Matrix equation for equilibrium P P = Remultiply this equation by A - to obtain P P = - - :. P = and P = (c) The augmented matrix of the system of equations is given by :. X = ; X = ; X = 9 Page of

9 QUANTITATIVE TOOLS IN BUSINESS NOV. SOLUTION (a) (b) A Chain Float is the spare time associate with a chain of non-critical activities. Free Float is the amount of time an activity can be delayed without affecting the commencement of a subsequent activity of its earliest start time, but may affect float of a previous activity. An independent float is the delay when all preceding activities are completed as late as possible and all succeeding activities completed as early as possible. Independent float = (EET LST - D) Free float = (EET - EST - D) (c) (i) 9 9 \\ // (ii) The critical path from the diagram is The diagram of the project is days (iii) Activity Total float FREE Float 9 (LET EST D) ( -) ( ) ( ) ( ) Page 9 of (EET -EPT D) ( ) ( ) ( ) ( ) Independent Float (EET LST D) ( -) ( ) ( ) ( -)

10 QUANTITATIVE TOOLS IN BUSINESS NOV. (v) Activities that have Total Float = free float Independent Float = are Critical path activities SOLUTION (a) Year Qtr Revenue Sun in qtrs Sum of qtrs 9 Trend Actual- Trend Actual Yield (b) Year Q Q Q 9 Total Average Adj Seasonal Variation (c) A reasonable trendline is drawn on the graph sheet. From the trendline Page of

11 QUANTITATIVE TOOLS IN BUSINESS NOV. th Quarter of, trend =. st Quarter of, trend =. nd Quarter of, trend =. rd Quarter of, trend =. Gradient =.9 =. - Estimated data value = Forecast trend value + appropriate seasonal variation value th Quarter of estimated value =. =. st Quarter of estimated value =. =. nd Quarter of estimated value =. + =. rd Quarter of estimated value =. + =. SOLUTION Let denote the occupancy rate of Hotel A. Then the average daily rate of occupancy is given by the expected value of X. E(x) = (.) (.9) + (.) (.) + (.9) (.) + (.9) (.) + (-.) (.) = =. Let Y denote the occupancy rate of the Hotel B. Then the average daily rate of occupancy is given by the expected value Y. E(Y) = (.) (.) + (.) (.) + (.) (.) + (.9 (.) + (.9) (.9) + (.) (.) = =. The average number of rooms occupied per day at the Hotel B is (. () = 9. The expected daily profit at Hotel A is given by (.) () = or GHS The expected daily profit at Hotel B is given by (9.) (9) = or GHS From these results we conclude that the investors should purchase Hotel A which is expected to yield a higher daily profit. Page of