James Beerd bakes cheesecakes and Black Forest cakes. During any month, he can bake at most 65 cakes. The costs per cake and the demands for cakes, which must be met on time, are listed in Table 33. Itcosts500 to hold a cheesecake, and 40¢ to hold a Black Forest cake, in inventory for a month. Formulate an LP to minimize the total cost of meeting the next three months’ demands. Table 33: $$ \begin{matrix}\text{ } & \text{Month 1} & \text{ } & \text{Month 2} & \text{ } & \text{Month 3} & \text{ }\\\text{Item} & \text{Demand} & \text{Cost/Cake }($) & \text{Demand} & \text{Cost/Cake }($) & \text{Demand} & \text{Cost/Cake }($) \\\text{Cheesecake} & \text{40} & \text{3.00} & \text{30} & \text{3.40} & \text{20} & \text{3.80}\\\text{Black Forest} & \text{20} & \text{2.50} & \text{30} & \text{2.80} & \text{10} & \text{3.40}\\\end{matrix} $$

A company faces the following demands during the next three periods: period 1, 20 units; period 2, 10 units; period 3, 15 units. The unit production cost during each period is as follows: period 1—$13; period 2—$14; period 3—$15. A holding cost of$2 per unit is assessed against each period’s ending inventory. At the beginning of period 1, the company has 5 units on hand. In reality, not all goods produced during a month can be used to meet the current month’s demand. To model this fact, we assume that only one half of the goods produced during a period can be used to meet the current period’s demands. Formulate an LP to minimize the cost of meeting the demand for the next three periods. (Hint: Constraints such as i1 = x1 + 5 – 20 are certainly needed. Unlike our example, however, the constraint $i 1 \geq 0$ will not ensure that period 1’s demand is met. For example, if x1 = 20, then $i 1 \geq 0$ will hold, bur because only $\frac{1}{2}(20)=10$ units of period 1 production can be used to meet period 1’s demand, x1 = 20 would not be feasible. Try to think of a type of constraint that will ensure that what is available to meet each period’s demand is at least as large as that period’s demand.)

Each year, Paynothing Shoes faces demands (which must be met on time) for pairs of shoes as shown in Table 37. Workers work three consecutive quarters and then receive one quarter off. For example, a worker may work during quarters 3 and 4 of one year and quarter 1 of the next year. During a quarter in which a worker works, he or she can produce up to 50 pairs of shoes. Each worker is paid $500 per quarter. At the end of each quarter, a holding cost of$50 per pair of shoes is assessed. Formulate an LP that can be used to minimize the cost per year (labor + holding) of meeting the demands for shoes. To simplify matters, assume that at the end of each year, the ending inventory is zero. (Hint: It is allowable to assume that a given worker will get the same quarter off during each year.) Table 37:

$$\begin{matrix} \text{Quarter 1} & \text{Quarter 2} & \text{Quarter 3} & \text{Quarter 4}\\ \text{600} & \text{300} & \text{800} & \text{100}\\\end{matrix}$$

Discuss how the vector equation of a line can be used to model the motion of a point that is moving with constant velocity in 3-space.

Bullco blends silicon and nitrogen to produce two types of fertilizers. Fertilizer 1 must be at least 40% nitrogen and sells for $70/lb. Fertilizer 2 must be at least 70% silicon and sells for$40/lb. Bullco can purchase up to 80 lb of nitrogen at $15/lb and up to 100 lb of silicon at$10/lb. Assuming that all fertilizer produced can be sold, formulate an LP to help Bullco maximize profits.

Jordan and Emery are rewriting the vertex form of the quadratic function

$$y = 2(x - 4)^2+5$$

in the form

$$y = ax^2 + bx + c.$$

Jordan

$$\left.\begin{aligned} y & = 2 ( x - 4 ) ^ { 2 } + 5 \\ & = ( 2 x - 8 ) ^ { 2 } + 5 \\ & = 4 x ^ { 2 } - 32 x + 64 + 5 \\ & = 4 x ^ { 2 } - 32 x + 69 \end{aligned} \right.$$

Emery

$$\left.\begin{aligned} y & = 2 ( x - 4 ) ^ { 2 } + 5 \\ & = 2 \left( x ^ { 2 } - 16 \right) + 5 \\ & = 2 x ^ { 2 } - 32 + 5 \\ & = 2 x ^ { 2 } - 27 \end{aligned} \right.$$

Communicate Precisely Did Jordan rewrite the equation correctly? Did Emery?

A rod of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the temperature u(x, t). The left end is at temperature $\sin (\pi t / L)$, the right end is held at zero, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at temperature zero. The initial temperature is $f(x)$ throughout.

How did the Renaissance emerge, and how did it change Europe?

For each of the following pairs of statements, use Modus Ponens or Modus Tollens to fill in the blank line so that a valid argument is presented. a). If Janice has trouble starting her car, then her daughter Angela will check Janice’s spark plugs. Janice had trouble starting her car. b) If Brady solved the first problem correctly, then the answer he obtained is 137. Brady’s answer to the first problem is not 137. c) If this is a repeat-until loop, then the body of this loop is executed at least once. The body of the loop is executed at least once. d) If Tim plays basketball in the afternoon, then he will not watch television in the evening. Tim didn’t play basketball in the afternoon.

A Las Vegas supermarket bakery must decide how many wedding cakes to prepare for the upcoming weekend. Cakes cost $33 each to make, and they sell for$60 each. Unsold cakes are reduced to half-price on Monday, and typically one-third of those are sold. Any that remain are donated to a nearby senior center. Analysis of recent demand resulted in the following table:

Demand 0 1 2 3 Probability .15 .35 .30 .20

How many cakes should be prepared to maximize expected profit? a. Use the ratio method. b. Use the tabular method

Suppose that r, the annual interest rate, is 0.20. and that all money in the bank cams 20% interest each year (that is, after being in the bank for one year, $1 will increase to$1.20). If we place $100 in the bank fix one year, what is the NPV of this transaction?

A can of cola has 41 grams of sugar in each 12-ounce can. If you drink nine cans over the span of a week, how many pounds of sugar have you consumed during that week from the soda?

A customer requires during the next four months, respectively, 50, 65, 100, and 70 units of a commodity (no backlogging is allowed). Production costs are $5,$8, $4, and$7 per unit during these months. The storage cost from one month to the next is $2 per unit (assessed on ending inventory). It is estimated that each unit on hand at the end of month 4 could be sold for$6. Formulate an LP that will minimize the net cost incurred in meeting the demands of the next four months.

Finco must determine how much investment and debt to undertake during the next year. Each dollar invested reduces the NPV of the company by 10¢, and each dollar of debt increases the NPV by 50¢ (due to deductibility of interest payments). Finco can invest at most $1 million during the coming year. Debt can be at most 40% of investment. Finco now has$800,000 in cash available. All investment must be paid for from current cash or borrowed money. Set up an LP whose solution will tell Finco how to maximize its NPV. Then graphically solve the LP.