MAT1050 / Dr. Khoule Exam 3 1. The total revenue function for a product is given by same product is given by R 250x C 1600 350x x 2 , and the total cost function for this , where R and C are each measured in thousands of dollars and x is the number of units produced and sold. a) Form the profit function for this product from the given two equations. b) What is the profit when 60 units are produced and sold? c) How many units must be sold to break even on this product? 2. Solve the quadratic equation 3. Sales Decay 5 x 2 7 x 6 At the end of an advertising campaign, weekly sales declined according to the equation dollars, where x is the number of weeks after the end of the ad campaign. y 10,000(3 0.04 x ) a) Determine the sale at the end of the ad campaign. b) How many weeks will take for sales to reach $4,500? c) Will sales ever reach $0? 4. The weekly total cost function for producing a copy machine is C(x) = 6100 + 102x, where x is the number of copy machine produced per week. a) Form the weekly average cost function for this product. b) Find the weekly cost for the production of 200 copy machines. c) Find the average cost for the production of 200 copy machines. d) Explain how the difference between Parts (b) and (c). 5. The demand for a brand of clock radio is given by p + 2q = 320, and the supply for these radios is given by p - 8q = 20, where p is the price and q is the quantity demanded at price p. Solve the system to find the price at which the quantity demanded equals the quantity supplied and the equilibrium quantity. 6. Given the linear function 6 x 4 y 24 Find a) the slope b) the x-and y-intercept . a) graph the function. 7. Find the equation of the line that passes through the points (-1, 3) and (2, 6). 8. Find the equation of the line that passes through the points (2, 3) and (2, 6). 9. Find the equation of the line that passes through the points (2, 3) and (4, 3). 10. Find the equation of the line that passes through the points (3, 2) and perpendicular to the line 2x + 6y =1 11. Use . f ( x) x 4 5 x 3 3 x 4 a. Use long division to divide by . f (x) ( x 2) b. Is (x + 1) a factor of f (x) ? 12. Solve the system of equation for x, y and z: 2 x 3 y z 1 2 x 4 y 3 z 9 x 3 y 5 z 5 13. The sum of a student's three scores is 220. If the first is 15 points more than the second, and the sum of the first two is 31 more than twice the third, then what was the first score? Solve algebraically. 14. Write in logarithmic form: 43 64 . The demand for a brand of clock radio is given by p + 2q = 320 and the supply for these radios is given by p - 8q = 20 The total revenue function for a product is given by R = 250x and the total cost function for this Form the profit function for this product from the given two equations 1 a profit revenue cost 250 x 1600 350 x x 2 p x 250 x 1600 350 x x 2 p x 1600 100 x x 2 b Plug x 60 in p x 1600 100 x x 2 p 60 1600 100 60 60 8000 2 c 250 x 1600 350 x x 2 250 x 1600 350 x x 2 0 1600 100 x x 2 0 x 80, 20 2 Given that 5x 2 7 x 6 5x2 7 x 6 0 x 7 49 120 2 5 7 169 10 7 13 x 10 7 13 7 13 x ,x 10 10 6 20 x ,x 10 10 3 x , x 2 5 x 3 a The equation y 10000 30.04 x where x is the number of weeks after the end of the campaign So, plug x 6 in y 10000 30.04 x y 10000 30.04 6 7682.293564 7682.3 b How many weeks will take for sales to reach $4,500 4,500 10000 30.04 x 4,500 30.04 x 10000 0.45 30.04 x x 18.171 c Will sales ever reach $0 No,because y 10000 30.04 0 y 10000 4 Given that C x 6100 102 x a Average cost function: C x 6100 102 x 6100 102 x x x b 6100 Plug x 200 in 102 x 6100 102 200 61 102 2 265 2 c The average cost for the production of 200 copy machines is d 265 2x Part b is only weekly cost for the production of 200 copy machines Part c is The average cost for the production of 200 copy machines So that the difference of b and c is 265 265 265 1 1 2x 2 2 x 265 1 x 2 x 5 Given that p 2q 320, p - 8q 20 Solve the above p 2q 320, p - 8q 20 10q 280 q 28 Plug q 28 in p - 8q 20 p - 8 28 20 p 244 The quantity demanded equals the quantity supplied and the equilibrium quantity at p 244 and q 28 6 Given that 6 x 4 y 24 4 y 6 x 24 3 y x6 2 a Slope is m b 3 2 6 x 4 y 24 1 1 x y 1 4 6 x intercept is 4 y intercept is 6 c The graph of the 6 x 4 y 24 as follows 7 Slope of 1,3 and 2, 6 3 1 3 So, the equation y 3 1 x 1 m y x4 8 Slope of 2,3 and 2, 6 m is undefined means that the line is vertical. vertical line passing through 2,3 is x 2 So, the equation is x 2 9 Slope of 2,3 and 4,3 m is 0 So, the equation y 3 0 x 2 That is y 3 10 Given that 2 x 6 y 1 6 y 2 x 1 x 1 y 3 6 perpendicular slope is 3 So the equation is y 2 3 x 3 y 3x 7 11 f x x 4 5 x 3 3x 4 x 4 5 x 3 3x 4 x 4 2 x3 3x3 3x 4 3x3 6 x 2 x 2 6 x 2 3 x 4 x 3 3x 2 6 x 15 6 x 2 12 x 15 x 4 15 x 30 34 b Plug x 1 in f x x 4 5 x 3 3x 4 f 1 1 5 1 3 1 4 4 3 1 5 3 4 0 So, x 1 is not a factor of f x 12 2x 3y z 1 2 x 4 y 3z 9 x 3 y 5z 5 2 3 1 1 2 4 3 9 1 3 5 5 Add -1 times row 1 to row 2 Add -1/2 times row 1 to row 3 2 3 1 1 0 1 2 8 3 9 9 0 2 2 2 Add -3/2 times row 2 to row 3 2 3 1 1 8 0 1 2 3 15 0 0 2 2 2x 3y z 1 y 2z 8 3 15 z 2 2 x 24, y 18, z 5 13 Let x be the first score Then x 15 The second score Then the sum of the first two x x 15 is 31 more than twice the third x x 15 31 2 x 46 x 23 2 2 Equation x x 15 x 23 220 x 85 14 43 64 log 4 64 3 The demand for a brand of clock radio is given by p + 2q = 320 and the supply for these radios is given by p - 8q = 20 The total revenue function for a product is given by R = 250x and the total cost function for this Form the profit function for this product from the given two equations 1 a profit revenue cost 250 x 1600 350 x x 2 p x 250 x 1600 350 x x 2 p x 1600 100 x x 2 b Plug x 60 in p x 1600 100 x x 2 p 60 1600 100 60 60 8000 2 c 250 x 1600 350 x x 2 250 x 1600 350 x x 2 0 1600 100 x x 2 0 x 80, 20 2 Given that 5x 2 7 x 6 5x2 7 x 6 0 x 7 49 120 2 5 7 169 10 7 13 x 10 7 13 7 13 x ,x 10 10 6 20 x ,x 10 10 3 x , x 2 5 x 3 a The equation y 10000 30.04 x where x is the number of weeks after the end of the campaign So, plug x 6 in y 10000 30.04 x 0.04 6 y 10000 3 7682.293564 7682.3 b How many weeks will take for sales to reach $4,500 4,500 10000 30.04 x 4,500 30.04 x 10000 0.45 30.04 x x 18.171 c Will sales ever reach $0 No,because y 10000 30.04 0 y 10000 4 Given that C x 6100 102 x a Average cost function: C x x b 6100 102 x 6100 102 x x 6100 Plug x 200 in 102 x 6100 102 200 61 102 2 265 2 c The average cost for the production of 200 copy machines is d 265 2x Part b is only weekly cost for the production of 200 copy machines Part c is The average cost for the production of 200 copy machines So that the difference of b and c is 265 265 265 1 1 2x 2 2 x 265 1 x 2 x 5 Given that p 2q 320, p - 8q 20 Solve the above p 2q 320, p - 8q 20 10q 280 q 28 Plug q 28 in p - 8q 20 p - 8 28 20 p 244 The quantity demanded equals the quantity supplied and the equilibrium quantity at p 244 and q 28 6 Given that 6 x 4 y 24 4 y 6 x 24 y a 3 x6 2 Slope is m b 3 2 6 x 4 y 24 1 1 x y 1 4 6 x intercept is 4 y intercept is 6 c The graph of the 6 x 4 y 24 as follows 7 Slope of 1,3 and 2, 6 3 1 3 So, the equation y 3 1 x 1 m y x4 8 Slope of 2,3 and 2, 6 m is undefined means that the line is vertical. vertical line passing through 2,3 is x 2 So, the equation is x 2 9 Slope of 2,3 and 4,3 m is 0 So, the equation y 3 0 x 2 That is y 3 10 Given that 2 x 6 y 1 6 y 2 x 1 x 1 y 3 6 perpendicular slope is 3 So the equation is y 2 3 x 3 y 3x 7 11 f x x 4 5 x3 3x 4 x 4 5 x3 3x 4 x 4 2 x3 3x3 3x 4 3x3 6 x 2 x 2 6 x 2 3 x 4 x 3 3 x 2 6 x 15 6 x 2 12 x 15 x 4 15 x 30 34 b Plug x 1 in f x x 4 5 x3 3x 4 f 1 1 5 1 3 1 4 4 3 1 5 3 4 0 So, x 1 is not a factor of f x 12 2x 3y z 1 2 x 4 y 3z 9 x 3 y 5z 5 2 3 1 1 2 4 3 9 1 3 5 5 Add -1 times row 1 to row 2 Add -1/2 times row 1 to row 3 2 3 1 1 0 1 2 8 3 9 9 0 2 2 2 Add -3/2 times row 2 to row 3 2 3 1 1 8 0 1 2 3 15 0 0 2 2 2x 3y z 1 y 2z 8 3 15 z 2 2 x 24, y 18, z 5 13 Let x be the first score Then x 15 The second score Then the sum of the first two x x 15 is 31 more than twice the third x x 15 31 2 x 46 x 23 2 2 Equation x x 15 x 23 220 x 85 14 43 64 log 4 64 3 The demand for a brand of clock radio is given by p + 2q = 320 and the supply for these radios is given by p - 8q = 20 The total revenue function for a product is given by R = 250x and the total cost function for this Form the profit function for this product from the given two equations 1 a profit revenue cost 250 x 1600 350 x x 2 p x 250 x 1600 350 x x 2 p x 1600 100 x x 2 b Plug x 60 in p x 1600 100 x x 2 p 60 1600 100 60 60 8000 2 c 250 x 1600 350 x x 2 250 x 1600 350 x x 2 0 1600 100 x x 2 0 x 80, 20 2 Given that 5x 2 7 x 6 5x2 7 x 6 0 x 7 49 120 2 5 7 169 10 7 13 x 10 7 13 7 13 x ,x 10 10 6 20 x ,x 10 10 3 x , x 2 5 x 3 a The equation y 10000 30.04 x where x is the number of weeks after the end of the campaign So, plug x 6 in y 10000 30.04 x y 10000 30.04 6 7682.293564 7682.3 b How many weeks will take for sales to reach $4,500 4,500 10000 30.04 x 4,500 30.04 x 10000 0.45 30.04 x x 18.171 c Will sales ever reach $0 No,because y 10000 30.04 0 y 10000 4 Given that C x 6100 102 x a Average cost function: C x 6100 102 x 6100 102 x x x b 6100 Plug x 200 in 102 x 6100 102 200 61 102 2 265 2 c The average cost for the production of 200 copy machines is d 265 2x Part b is only weekly cost for the production of 200 copy machines Part c is The average cost for the production of 200 copy machines So that the difference of b and c is 265 265 265 1 1 2x 2 2 x 265 1 x 2 x 5 Given that p 2q 320, p - 8q 20 Solve the above p 2q 320, p - 8q 20 10q 280 q 28 Plug q 28 in p - 8q 20 p - 8 28 20 p 244 The quantity demanded equals the quantity supplied and the equilibrium quantity at p 244 and q 28 6 Given that 6 x 4 y 24 4 y 6 x 24 3 y x6 2 a Slope is m b 3 2 6 x 4 y 24 1 1 x y 1 4 6 x intercept is 4 y intercept is 6 c The graph of the 6 x 4 y 24 as follows 7 Slope of 1,3 and 2, 6 3 1 3 So, the equation y 3 1 x 1 m y x4 8 Slope of 2,3 and 2, 6 m is undefined means that the line is vertical. vertical line passing through 2,3 is x 2 So, the equation is x 2 9 Slope of 2,3 and 4,3 m is 0 So, the equation y 3 0 x 2 That is y 3 10 Given that 2 x 6 y 1 6 y 2 x 1 x 1 y 3 6 perpendicular slope is 3 So the equation is y 2 3 x 3 y 3x 7 11 f x x 4 5 x 3 3x 4 x 4 5 x 3 3x 4 x 4 2 x3 3x3 3x 4 3x3 6 x 2 x 2 6 x 2 3 x 4 x 3 3x 2 6 x 15 6 x 2 12 x 15 x 4 15 x 30 34 b Plug x 1 in f x x 4 5 x 3 3x 4 f 1 1 5 1 3 1 4 4 3 1 5 3 4 0 So, x 1 is not a factor of f x 12 2x 3y z 1 2 x 4 y 3z 9 x 3 y 5z 5 2 3 1 1 2 4 3 9 1 3 5 5 Add -1 times row 1 to row 2 Add -1/2 times row 1 to row 3 2 3 1 1 0 1 2 8 3 9 9 0 2 2 2 Add -3/2 times row 2 to row 3 2 3 1 1 8 0 1 2 3 15 0 0 2 2 2x 3y z 1 y 2z 8 3 15 z 2 2 x 24, y 18, z 5 13 Let x be the first score Then x 15 The second score Then the sum of the first two x x 15 is 31 more than twice the third x x 15 31 2 x 46 x 23 2 2 Equation x x 15 x 23 220 x 85 14 43 64 log 4 64 3 The demand for a brand of clock radio is given by p + 2q = 320 and the supply for these radios is given by p - 8q = 20 The total revenue function for a product is given by R = 250x and the total cost function for this Form the profit function for this product from the given two equations 1 a profit revenue cost 250 x 1600 350 x x 2 p x 250 x 1600 350 x x 2 p x 1600 100 x x 2 b Plug x 60 in p x 1600 100 x x 2 p 60 1600 100 60 60 8000 2 c 250 x 1600 350 x x 2 250 x 1600 350 x x 2 0 1600 100 x x 2 0 x 80, 20 2 Given that 5x 2 7 x 6 5x2 7 x 6 0 x 7 49 120 2 5 7 169 10 7 13 x 10 7 13 7 13 x ,x 10 10 6 20 x ,x 10 10 3 x , x 2 5 x 3 a The equation y 10000 30.04 x where x is the number of weeks after the end of the campaign So, plug x 6 in y 10000 30.04 x 0.04 6 y 10000 3 7682.293564 7682.3 b How many weeks will take for sales to reach $4,500 4,500 10000 30.04 x 4,500 30.04 x 10000 0.45 30.04 x x 18.171 c Will sales ever reach $0 No,because y 10000 30.04 0 y 10000 4 Given that C x 6100 102 x a Average cost function: C x x b 6100 102 x 6100 102 x x 6100 Plug x 200 in 102 x 6100 102 200 61 102 2 265 2 c The average cost for the production of 200 copy machines is d 265 2x Part b is only weekly cost for the production of 200 copy machines Part c is The average cost for the production of 200 copy machines So that the difference of b and c is 265 265 265 1 1 2x 2 2 x 265 1 x 2 x 5 Given that p 2q 320, p - 8q 20 Solve the above p 2q 320, p - 8q 20 10q 280 q 28 Plug q 28 in p - 8q 20 p - 8 28 20 p 244 The quantity demanded equals the quantity supplied and the equilibrium quantity at p 244 and q 28 6 Given that 6 x 4 y 24 4 y 6 x 24 y a 3 x6 2 Slope is m b 3 2 6 x 4 y 24 1 1 x y 1 4 6 x intercept is 4 y intercept is 6 c The graph of the 6 x 4 y 24 as follows 7 Slope of 1,3 and 2, 6 3 1 3 So, the equation y 3 1 x 1 m y x4 8 Slope of 2,3 and 2, 6 m is undefined means that the line is vertical. vertical line passing through 2,3 is x 2 So, the equation is x 2 9 Slope of 2,3 and 4,3 m is 0 So, the equation y 3 0 x 2 That is y 3 10 Given that 2 x 6 y 1 6 y 2 x 1 x 1 y 3 6 perpendicular slope is 3 So the equation is y 2 3 x 3 y 3x 7 11 f x x 4 5 x3 3x 4 x 4 5 x3 3x 4 x 4 2 x3 3x3 3x 4 3x3 6 x 2 x 2 6 x 2 3 x 4 x 3 3 x 2 6 x 15 6 x 2 12 x 15 x 4 15 x 30 34 b Plug x 1 in f x x 4 5 x3 3x 4 f 1 1 5 1 3 1 4 4 3 1 5 3 4 0 So, x 1 is not a factor of f x 12 2x 3y z 1 2 x 4 y 3z 9 x 3 y 5z 5 2 3 1 1 2 4 3 9 1 3 5 5 Add -1 times row 1 to row 2 Add -1/2 times row 1 to row 3 2 3 1 1 0 1 2 8 3 9 9 0 2 2 2 Add -3/2 times row 2 to row 3 2 3 1 1 8 0 1 2 3 15 0 0 2 2 2x 3y z 1 y 2z 8 3 15 z 2 2 x 24, y 18, z 5 13 Let x be the first score Then x 15 The second score Then the sum of the first two x x 15 is 31 more than twice the third x x 15 31 2 x 46 x 23 2 2 Equation x x 15 x 23 220 x 85 14 43 64 log 4 64 3