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MATH 1100: QUANTITATIVE ANALYSIS TEST #1 (VERSION A) 1. By calculating the reduced form, find all solutions to x + 2y − z = 3 3x + y = 4 2x − y + z = 2. Solution: There are no solutions. 2. Using matrix inverses, solve: 4x − y = 1 x + 2y = 2 Solution: 4 −1 1 2 −1 = 2 9 − 19 1 9 4 9 and so x= 4 7 ,y = . 9 9 3. Suppose that $1000 is invested at 10% nominal interest rate, compounded continuously. (a) Write an equation describing the value of the investment after t years since the principal was invested. (b) Approximate the value of the investment after 30 years, using the approximation e ∼ 3. (c) How many years does it take for the investment to double in value? Hint: use the approximation ln(2) ∼ 0.7. Solution: (a) S = 1000e0.10t (b) S = 1000e0.10×30 = 1000e3 ∼ 1000 × 33 = 1000 × 27 = 27000. Date: September 3, 2001. 1 2 MATH 1100: QUANTITATIVE ANALYSIS TEST #1 (VERSION A) 300 200 100 0 100 200 300 400 A –100 Figure 1. Beside the two axes, there are two other constraints (c) 0.7 ln(2) = = 7 years. r 0.1 4. Your company makes two products, A’s and B’s. The A’s require 2 hours of work each to make, while the B’s require 4. You have 800 work hours available each day to make them. But you can only transport a total of 300 products (A’s and B’s together) in a day. A’s sell for $30, B’s for $40. Suppose you will always sell as many as you make. How many of each should you make to maximize sales revenue? Solution: See the picture in figure 1. The constraints are t= 2A + 4B ≤ 800 A + B ≤ 300 A≥0 B≥0 This gives the part of the upper right quadrant which is beneath both of the lines 2A + 4B = 800 and A + B = 300. The revenue is R = 30A + 40B. The maximum is at 200 A’s, 100 B’s.