A company must meet (on time) the following demands: quarter 1—30 units; quarter 2—20 units; quarter 3—40 units. Each quarter, up to 27 units can be produced with regular-time labor, at a cost of $40 per unit. During each quarter, an unlimited number of units can be produced with overtime labor, at a cost of$60 per unit. Of all units produced, 20% arc unsuitable and cannot be used to meet demand. Also, at the end of each quarter, 10% of all units on hand spoil and cannot be used to meet any future demands. After each quarter’s demand is satisfied and spoilage is accounted for, a cost of $ 15 per unit is assessed against the quarter’s ending inventory. Formulate an LP that can be used to minimize the total cost of meeting the next three quarters’ demands. Assume that 20 usable units are available at the beginning of quarter 1.

Donovan Enterprises produces electric mixers. During the next four quarters, the following demands for mixers must be met on time: quarter 1—4,000; quarter 2—2,000; quarter 3—3,000; quarter 4—10,000. Each of Donovan’s workers works three quarters of the year and gets one quarter off. Thus, a worker may work during quarters 1, 2, and 4 and get quarter 3 off. Each worker is paid $30,000 per year and (if working) can produce up to 500 mixers during a quarter. At the end of each quarter, Donovan incurs a holding cost of$30 per mixer on each mixer in inventory. Formulate an LP to help Donovan minimize the cost (labor and inventory) of meeting the next year’s demand (on time). At the beginning of quarter 1, 600 mixers are available.

After the accounts are adjusted and closed at the end of the fiscal year, Accounts Receivable has a balance of $\$298,150$ and Allowance for Doubtful Accounts has a balance of $\$31,200$. Describe how the accounts receivable and the allowance for doubtful accounts are reported on the balance sheet.

Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.

$$\left. \begin{array} { l } { y ^ { \prime } = y - \cos \frac { \pi } { 2 } x } \\ { \textbf{(a) } y ( 2 ) = 2 } \\ { \textbf{ (b) } y ( - 1 ) = 0 } \end{array} \right.$$

Graph each absolute value equation. y = |2x + 6|

Which objects are asymmetric (have no plane of symmetry): (a) a circular clock face; (b) a footballs; (c) a dime; (d) a brick; (e) a hammer; (f) a spring?

Televco produces TV picture tubes at three plants. Plant 1 can produce 50 tubes per week; plant 2, 100 tubes per week; and plant 3, 50 tubes per week. Tubes are shipped to three customers. The profit earned per tube depends on the site where the tube was produced and on the customer who purchases the tube (see Table 64). Customer 1 is willing to purchase as many as 80 tubes per week; customer 2, as many as 90; and customer 3, as many as 100. Televco wants to find a shipping and production plan that will maximize profits. a. Formulate a balanced transportation problem that can be used to maximize Televco’s profits b. Use the northwest corner method to find a bfs to the problem. c. Use the transportation simplex to find an optimal solution to the problem. TABLE 64:

$$\begin{matrix} \text{From} & \text{To (\$)}\\ \text{ } & \text{Customer 1} & \text{Customer 2} & \text{Customer 3}\\ \text{Plant 1} & \text{75} & \text{60} & \text{69}\\ \text{Plant 2} & \text{79} & \text{73} & \text{68}\\ \text{Plant 3} & \text{85} & \text{76} & \text{70}\\ \end{matrix}$$

Broker Steve Johnson is currently trying to maximize his profit in the bond market. Four bonds are available for purchase and sale, with the bid and ask pricc of each bond as shown in Table 40. Steve can buy up to 1,000 units of each bond at the ask price or sell up to 1,000 units of each bond at the bid price. During each of the next three years, the person who sells a bond will pay the owner of the bond the cash payments shown in Table 41. Steve’s goal is to maximize his revenue from selling bonds less his payment for buying bonds, subject to the constraint that after each year’s payments are received, his current cash position (due only to cash payments from bonds and not purchases or sale of bonds) is nonnegative. Assume that cash payments are discounted, with a payment of $1 one year from now being equivalent to a payment of 90¢ now. Formulate an LP to maximize net profit from buying and selling bonds, subject to the arbitrage constraints previously described. Why do you think we limit the number of units of each bond that can be bought or sold? Table 40:

$$\begin{matrix} \text{Bond} & \text{Bid Price} & \text{Ask Price}\\ \text{1} & \text{980} & \text{990}\\ \text{2} & \text{970} & \text{985}\\ \text{3} & \text{960} & \text{972}\\ \text{4} & \text{940} & \text{954}\\ \end{matrix}$$

Table 41:

$$\begin{matrix} \text{Year} & \text{Bond 1} & \text{Bond 2} & \text{Bond 3} & \text{Bond 4}\\ \text{1} & \text{100} & \text{80} & \text{70} & \text{60}\\ \text{2} & \text{110} & \text{90} & \text{80} & \text{50}\\ \text{3} & \text{1,100} & \text{1,120} & \text{1,090} & \text{1,110}\\ \end{matrix}$$

For each key term in the lesson, write a sentence explaining its significance.

Find a polynomial of the specified degree that has the given zeros. Degree 5; zeros -2, -1, 0, 1, 2

Solve the linear inequality. Express the solution using interval notation and graph the solution set.

$$- 6 \leq 3 x - 7 \leq 8$$

In which of these four ancient Mediterranean cultures— nomadic, Phoenician, Israelite, and Minoan— would you choose to live, if you could choose? Write an essay that explains your choice and why you would prefer that culture over the others.

Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. f(x)=

$$\tan ^ { - 1 } \left( 4 x ^ { 2 } \right)$$

In Problem, assume that before being shipped to Los Angeles or New York, all oil produced at the wells must be refined at either Galveston or Mobile. To refine 1,000 barrels of oil costs $12 at Mobile and$10 at Galveston. Assuming that both Mobile and Galveston have infinite refinery capacity, formulate a transshipment and balanced transportation model to minimize the daily cost of transporting and refining the oil requirements of Los Angeles and New York. Problem: Sunco Oil produces oil at two wells. Well 1 can produce as many as 150,000 barrels per day, and well 2 can produce as many as 200,000 barrels per day. It is possible to ship oil directly from the wells to Sunco’s customers in Los Angeles and New York. Alternatively, Sunco could transport oil to the ports of Mobile and Galveston and then ship it by tanker to New York or Los Angeles. Los Angeles requires 160,000 barrels per day, and New York requires 140,000 barrels per day. The costs of shipping 1,000 barrels between two points areshown in Table 61. Formulate a transshipment model (and equivalent transportation model) that could be used to minimize the transport costs in meeting the oil demands of Los Angeles and New York. Table 61:

$$\begin{matrix} \text{From} & \text{To (\$)}\\ \text{ } & \text{Well 1} & \text{Well 2} & \text{Mobile} & \text{Galveston} & \text{N.Y.} & \text{L.A.}\\ \text{Well 1} & \text{0} & \text{—} & \text{10} & \text{13} & \text{25} & \text{28}\\ \text{Well 2} & \text{—} & \text{0} & \text{15} & \text{12} & \text{26} & \text{25}\\ \text{Mobile} & \text{—} & \text{—} & \text{0} & \text{6} & \text{16} & \text{17}\\ \text{Galveston} & \text{—} & \text{—} & \text{6} & \text{0} & \text{14} & \text{16}\\ \text{N.Y.} & \text{—} & \text{—} & \text{—} & \text{—} & \text{0} & \text{15}\\ \text{L.A.} & \text{—} & \text{—} & \text{—} & \text{—} & \text{15} & \text{0}\\ \end{matrix}$$

Note: Dashes indicate shipments that are not allowed.