1. Formulate, but do not solve, the following two exercises as linear programming problems. (a) A company manufactures x number of plush cats, and y number of plush dogs. Each plush toy must go through cutting and assembly. A plush cat requires 5 minutes of cutting and 5 minutes of assembly. Each plush dog requires 8 minutes of cutting and 5 minutes of assembly. There is an hour and a half available for cutting and 95 minutes available for assembly each shift. A plush cat will sell for $4 and each plush dog sells for $5. How many of each plush toy should be produced in order to maximize the revenue? (b) A manufacturer produces two models of fireplace grates, x units of model A and y units of model B. To produce each model A requires 2 pounds of cast iron and 8 minutes of labor. To produce each model B grate requires 5 pounds of cast iron and 3 minutes of labor. The profit from each model A grate is $2.00 and the profit from each model B grate is $1.50. The manufacturer has 1100 pounds of cast iron and 25 hours available for labor each day. Due to a backlog in orders, the manufacturer must make at least 150 model A grates per day. How many of each model should the manufacturer make in order to maximize profit?
2. Matrix A is a 6 3 matrix, B is 3 8, and matrix C is 6 8. Use this information to find the sizes of the following matrices. (a) AB (b) BA (c) C T (d) A + C (e) B(C T )
3. Compute the matrix multiplication. 4 9 [ ] (a) 8 4 5 10 5 5 10 7 4 10 (b) [ 11 4x 7y 15 ] [ 3 6 5 5 ] (c) [ 3 6 5 5 ] [ 11 4x 7y 15 ]
4. (a) A business owner owns 2 gas stations at locations A and B. On a given Saturday, gas station A sold 1200 gallons of regular, 750 gallons of regular plus, and 650 gallons of premium gasoline. On the same day, gas station B sold 1100 gallons of regular, 850 gallons of regular plus, and 600 gallons of premium. Write the information in a matrix, with rows representing the location of the gas station and columns representing types of gas. (b) On the next day, station A sold 1250 gallons of regular, 825 gallons of regular plus, and 550 gallons of premium. Station B sold 1150 gallons of regular, 750 gallons of regular plus, and 750 gallons premium. Write this information as a matrix, similar to part (a). (c) Find a matrix representing the total sales of the two gas stations over the two day period. (d) Suppose that the prices for gasoline are $1.99 for a gallon of regular, $2.10 for a gallon of regular plus, and $2.50 for a gallon of premium. Write this information in a matrix, with rows representing types of gas, and columns representing price per gallon. (e) Multiply the matrices in parts (c) and (d) to find the amount of revenue the gas station owner made for each station that weekend.
5. John Peters (you know, the farmer), plans to plant two types of trees, peaches and oranges. He has 500 acres of land allotted for these crops. The cost of cultivating peaches is $42 per acre and the cost of an orange grove is $30 per acre. He has $16,800 available for this venture. Due to (oddly specific) regulations from the city council, he must plant three sevenths the number of acres of peach trees as orange trees. He used all his budget and all his available land for these crops. (a) Set up the system of equations needed to find the number of acres of each type of tree. (b) Solve the system to find the number of acres of each type of tree he planted. 6. Solve X + 3A T = B for X if matrix A = [ 5 5 3 5 ] [ 10 1 and B = 7 9 ].
7. Solve the following systems of equations: (a) 4x + y z = 4 8x + 2y = 8 + 2z (b) 2x 1 x 2 x 3 = 0 3x 1 + x 3 = 7 2x 2 5 x 1 2x 2 = 2x 3 (c) 2x + y 3z = 1 x y + 2z = 1 5x 2y + 3z = 6 1 2 8. Solve for w, x, y, and z in 3 10 3 x 1 y 1 2 1 4 2 2z + 1 = 2 7 w 0 1 3 1
9. Determine if the following matrices are in row reduced form. If they are not, use row operations to row reduce them. [ ] 1 1 3 (a) 0 0 0 (b) 1 0 1 3 0 1 0 4 0 0-1 6 10. For a certain commodity, the price-supply equation is 3x 11p + 61 = 0 and the pricedemand equation is 2x + 9p 90 = 0 where x is measured in thousands of units and p is price per unit in dollars. Find the equilibrium quantity and equilibrium price. Round to the nearest unit/cent if necessary.
11. The quantity demanded each week of a certain brand of cell phones is 250 when the unit price is $144. The quantity demanded each week increases to 1000 when the unit price decreases by $30. The suppliers are unwilling to manufacture any cell phones if the price is $61 or less. They will market 700 when the price is $105 per cell phone. Both supply and demand equations are known to be linear. (a) Find the price-supply and price-demand equations. (b) Graph the supply and demand equations. 150 100 50 200 400 600 800 (c) Find the equilibrium quantity and price. Round to the nearest unit/cent..
12. A manufacturer of portable cell phone chargers has a monthly fixed cost of $4,000 and a production cost of $6 per charger. The manufacturer makes $5500 in revenue when 500 chargers are sold. (a) Find the total cost equation. (b) Find the total revenue equation. (c) Graph the cost and revenue equations. 8000 6000 4000 2000 2000 4000 6000 8000 (d) Find the break even quantity for the manufacturer.
13. Find the equation of the vertical line passing through the point ( 7, 5). 14. A building costs $800,000 when new. After 10 years, the building is scrapped. The materials used in the building are sold for $600,000 after its lifespan. (a) If the depreciation for the building is linear, find the equation relating the value V, and the time t (in years). (b) Find the value of the building after 8 years. (c) At what point is the value of the building $750,000?