# Mat 540 Test 4

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160.131 Mathematics for Business (1) Test 1 4th of April, 2012 Formulas you may need Slopes (Gradients) The slope (or gradient) m of a curve measures the rate of change in the dependent variable (y), i.e. m = [pic] = [pic] = [pic] = [pic] Determining the Equation of a Line: If we know 2 distinct points (x1, y1) and (x2, y2) on the line: y – y1 = [pic](x – x1); if we know a point (x0, y0) and the slope m of the line: y – y0 = m(x – x0); if we know the slope m and the y-intercept (0, c) of the line: y = mx + c. multiplier = - [pic] Inverse Matrices for 2 × 2 matrices: If A = [pic] then A-1 = [pic][pic], provided ad – bc [pic] 0. Best Fit Curves and…show more content…
Department Product 1 Product 2 Product 3 Hours Available per week A 2 3.5 3 1200 B 3 2.5 2 1150 C 4 3 2 1400 (29 marks) Answer: Suppose we produce x units of Product 1, y units of Product 2, and z units of Product 3. Then, we expect that 2x + 3.5y + 3z = 1200, 3x + 2.5y + 2z = 1150, and 4x + 3y + 2z = 1400. We then need to solve this system. 3 marks i. Convert the given system to the matrix form: [pic][pic] = [pic] 2 marks ii. Form the augmented matrix: 2 3.5 3 : 1200 3 2.5 2 : 1150 2 marks 4 3 2 : 1400 If you are familiar with this stuff you can skip Steps 1 & 2. iii. Apply row operations to reduce the matrix to the upper triangular form: ( 2 3.5 3 1200 ( -1.5 -2 3 2.5 2 1150 ( ( 4 3 2 1400 ( ( -2.75 -2.5 -650 ( ( ( -4 -4 -1000 ( -0.6875 ( 0.25 37.5 12…show more content…
2 marks 2x + 3.5y + 3z = 1200 ( 2x + 3.5 ( 100 + 3 ( 150 = 1200 ( 2x = 400 ( x = 200. Hence the solution is x = 200, y = 100, z = 150. 2 marks This means if we produce 200 units of Product 1, 100 units of Product 2, and 150 units of product 3 then the products will exhaust the weekly capacities. 2 marks 4. (Exercise II.6, 2) The demand and supply functions for golf lessons at Takapuna Club are given as follows. Demand function: P = 200 – 5Q Supply function: P = 92 + 4Q. a. Calculate the equilibrium price and quantity. b. The government imposes a tax of \$9 per lesson. i. Write down the equation of supply function adjusted for tax. ii. Calculate the new equilibrium price and quantity with the tax imposed. iii. Outline the distribution of the tax, i.e. calculate the tax paid by the consumer and by the club. (20 marks) Answer: a. If we are only asked to find the equilibrium price and quantity (i.e. no sub-question (b)), the best way to do it is by substitution method. However, as we are also asked to do (b), we use inverse matrix to solve the system as the inverse can be used