ADM1305M-Midterm_1_Qs_231_Winter 2023 (1)

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1 ADM 1305 M: MATHEMATICS FOR BUSINESS Winter 2023 MIDTERM TEST I Instructor: Edward Olale 1 2 3 4 5 6 7 Total 8 15 15 20 12 10 20 100 PERSONAL ETHICS STATEMENT CONCERNING TELFER SCHOOL ASSIGNMENTS Individual Assignment: By signing this Statement, I am attesting to the fact that I have adhered to the fraud policies outlined in the Academic Regulations in the University’s Undergraduate Studies Calendar Academic Fraud Webpage. Feb 16 th 2023 Student Signature Date Student Last Name (print), First Name (print) Student Number

2 Question 1 (8 Points) The weekly demand of fresh oranges, as measured by number of oranges, for a household depends on the price of orange (in dollars) and can be approximated by q ( p ) = 20 − p . Suppose that the supply of oranges also depends on the price at can be approximated by q ( p ) = 10 + 9 p. a) What the natural domain of the price of orange (2 points) b) Determine the equilibrium price in dollars (3 points) c) Determine the equilibrium number of oranges (3 points) Question 2 (15 points)

3 Given the matrix A = [ 5 − 3 − 1 1 5 ] a) Show that A is invertible (4 points) b) Determine the matrix inverse A -1 (6 points) c) Solve the following system of equations: (5 points) { 5 x − 3 y = 10 − x + 1 5 y =− 6 Question 3 (15 points) A new Telsa manufacturing plant in Sothern Ontario plans to start manufacturing Telsa Model 3 based on the following cost of production: C ( x ) = 15 + 20 x + 2.4 x 2 x≥ 0

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4 Where x is the number of Model 3 cars manufactured in thousands; and C(x) is the cost of production in $ million. Telsa is estimating the price of each Model 3 at $45 thousand. a) Determine the profit function: ( 5 points ) b) Determine the number of Model 3 cars that will allow the plant to break-even. Justify your answer. ( 5 points ) c) Determine the number of Model 3 cars that maximize profit and the maximum profit. ( 5 points ) Question 4 (20 points) McDonald’s needs to know how the demand for its fast-food menu depends on the price it charges. During the first week of operations, Macdonald’s was charging $10.50 per meal and sold 615. Macdonald’s noticed that raising the price to $13.25, during the second week of operations, resulted in lowering demand to only 516 meals. a) Determine the demand function using the data above (6 points)

5 b) Use the model in part a) to estimate the demand if the unit price were raised to $15. (2 points) c) Given that each meal sold costs Macdonald’s C ( q ) = 6 q + 650 , determine which of the two prices ($14 or $15) would result in the largest weekly profit, and the size of that profit. (12 points) Question 5 (12 points) The shares of a newly listed company are doubling in value every 12 hours. If you bought $3,000 worth of the stock when it was first offered: a) Determine the appropriate model giving the value of your investment at each hour of the day of initial trading. (6 points)

6 b) How much will your shares be worth at the end of the first day of trading (i.e.,24 hours later)? (2 points) c) How long will it take for your initial investment to triple its value. Round your answer to two decimal points. Justify. (4 points) Question 6 (10 points) The Horizons S&P/TSX Capped Energy Indexed fund was tumbling by 16.5% (per annum) such that: A ( t ) = A 0 ( 1 − 0.165 12 ) 12 t Where A 0 is the (unknown) initially invested amount and t is the time of investment in years. Anticipating the falling trend to last much longer, a skeptical investor decides to withdraw completely should the initial investment drop by a quarter. Determine how long this investor should wait before implementing his investment plan? (Round to the nearest two decimal)

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7 Question 7 (20 points) A farmer needs to purchase three types of fertilizer (x, y, z) to apply into her wheat farm. However, her extension officer has advised that the number bags of each type of fertilizer must meet the system of equations below to obtain maximum yield per acre (i.e., the combination of x, y and z should be applied to each acre). Use Gauss-Jordan row reduction to solve the given systems of equation (obtain the amounts bags of x, y and z). Show your work: x − 1 2 y = 6 1 3 x + 1 3 y + 1 3 z = 4 1 2 x − 1 2 z = 2

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