ACCOUNTING AND FINANCIAL ENGINEERING - EXAMPLE 34.23 : An engineering company would like to produce 10000 units of control instruments. Let selling price per unit = $10, variable cost per unit = $2, overall fixed cost = $30000. (a) Calculate the income obtained when all units are sold out successfully. (b) Find the overall cost of production. (c) Calculate the percentage of gross profit obtained based on the answers in (a) and (b). (d) Find the minimum units that need to be sold out successfully in order to prevent losses. (e) How many minimum units of instruments that need to be produced in order to prevent losses, if all units produced are sold out successfully?
ACCOUNTING AND FINANCIAL ENGINEERING - ANSWER 34.23 : (a) Income = 10000 units x $10 / unit = $100000. (b) Overall cost of production = overall variable cost + overall fixed cost = 10000 units x $2 / unit + $30000 = $50000. (c) Percentage of gross profit = 100 x [ answer (a) - answer (b) ] / [ answer (a) ] = 100 x ($100000 - $50000) / ($100000) = 50 %. (d) Let N units x $10 / unit = $50000 = answer (b) where N = units successfully sold = $50000 / ($10 / unit) = 5000 units. (e) Let contribution margin = (selling price - variable cost) / unit = $ (10 - 2) = $8. M x (contribution margin) = overall fixed cost = $30000. M = ($30000) / ($8) = 3750 units need to be produced at minimum. The answer is given by Kang Chuen Tat; PO Box 6263, Dandenong, Victoria VIC 3175, Australia; SMS +61405421706; chuentat@hotmail.com; http://kangchuentat.wordpress.com.
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Question 56 - The kinetic behavior of an enzyme could be described using Michalis - Menten equation : Vo = Vmax [S] / (Km + [S]). Derive this equation from [ES] = [E]total [S] / (Km + [S]), Vmax = Kcat [E]total, Vo = Kcat [ES].
QUANTUM COMPUTING - EXAMPLE 32.3 : A system of linear congruences consists of 3 equations : X ≡ 1 (mod 3), X ≡ 3 (mod 5), X ≡ 4 (mod 6). X has positive values. (a) List the values of these equations from 1 to 35. Then find the minimum value of X. (b)(i) Find the least common multiple (LCM) of b = 3, 5 and 6 where X ≡ a (mod b). (ii) If b - a has the same value of all equations above, then X + (b - a) is divisible by LCM. Find the value of minimum value of X via LCM division.
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Question 33 – By using Excel program either on laptop or desktop PC, solve the differential equation dy / dx = -2y + x + 4 with h = 0.005, initial values : x = 0, y = 1. The 4th order Runge-Kutta method provides : y(N + 1) = y(N) + (1/6) (k1 + 2k2 +2k3 + k4), k1 = h [ -2y(N) + x(N) + 4 ], k2 = h { -2 [ y(N) + k1 / 2 ] + x(N) + h / 2 + 4 }, k3 = h { -2 [ y(N) + k2 / 2 ] + x(N) + h / 2 + 4 }, k4 = h { -2 [ y(N) + k3 ] + x(N) + h + 4 }. What is the value of y at x = 0.5?
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PROCESS CONTROL - EXAMPLE 6.3 : The differential equation is 3 dy / dt + 2y = 1 with y(0) = 1. (a) The Laplace transformation, L for given terms are : L (dy / dt) = sY(s) - y(0), L(y) = Y(s), L(1) = 1 / s. Use such transformation to find Y(s). (b) The initial value theorem states that : When t approaches 0 for a function of y(t), it is equal to a function of sY(s) when s approaches infinity. Use the initial value theorem as a check to the answer found in part (a).
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