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.

During the next three months Airco must meet (on time) the following demands for air conditioners: month 1, 300. month 2, 400; month 3, 500. Air conditioners can be produced in either New York or Los Angeles. It takes 1.5 hours of skilled labor to produce an air conditioner in Los Angeles, and 2 hours in New York. It costs $400 to produce an air conditioner in Los Angeles, and$350 in New York. During each month, each city has 420 hours of skilled labor available. It costs $100 to hold an air conditioner in inventory for a month. At the beginning of month 1, Airco has 200 air conditioners in stock. Formulate an LP whose solution will tell Airco how to minimize the cost of meeting air conditioner demands for the next three months.

You have been put in charge of the Melrose oil refinery. The refinery produces gas and heating oil from crude oil. Gas sells for $8 per barrel and must have an average “grade level” of at least 9. Healing oil sells for$6 a barrel and must have an average grade level of at least 7. At most, 2,000 barrels of gas and 600 barrels of heating oil can be sold. Incoming crude can be processed by one of three methods. The per barrel yield and per barrel cost of each processing method are shown in Table 32. For example, if we refine 1 barrel of incoming crude by method 1, it costs us $3.40 and yields .2 barrels of grade 6, .2 barrels of grade 8, and .6 barrels of grade 10. Before being processed into gas and heating oil, processed grades 6 and 8 may be sent through the catalytic cracker to improve their quality. For$1.30 per barrel, a barrel of grade 6 may be “cracked” into a barrel of grade 8. For $2 per barrel, a barrel of grade 8 may be cracked into a barrel of grade 10. Any leftover processed or cracked oil that cannot be used for heating oil or gas must be disposed of at a cost of$0.20 per barrel. Determine how to maximize the refinery’s profit. Table 32: $$ \begin{matrix} \text{Method} & \text{Grade 6} & \text{Grade 8} & \text{Grade 10} & \text{Cost }($)\\ \text{1} & \text{.2} & \text{.2} & \text{.6} & \text{3.40}\\ \text{2} & \text{.3} & \text{.3} & \text{.4} & \text{3.00}\\ \text{3} & \text{.4} & \text{.4} & \text{.2} & \text{2.60}\\ \end{matrix} $$

Let U={p, q, r, s, t}, D={p, r, s, t}, E={q, s}, F={p, t}, and G={s}. Determine whether each statement is true or false. D has exactly eight subsets and seven proper subsets.

A population of bacteria triples every week in a laboratory. The number of bacteria is modeled by the expression

$$900\times 3^x,$$

where x is the number of weeks after a scientist measures the population size. When x = −2, what does the value of the expression represent?

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:

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

Factor completely, or state that the polynomial is prime.

$$x^3-4x$$

During the next two months, General Cars must meet (on time) the following demands for trucks and cars: month 1—400 trucks, 800 cars; month 2—300 trucks, 300 cars. During each month, at most 1,000 vehicles can be produced. Each truck uses 2 tons of steel, and each car uses 1 ton of steel. During month 1, steel costs $400 per ton; during month 2, steel costs$600 per ton. At most, 1,500 tons of steel may be purchased each month (steel may only be used during the month in which it is purchased). At the beginning of month 1, 100 trucks and 200 cars are in inventory. At the end of each month, a holding cost of $150 per vehicle is assessed. Each car gets 20 mpg, and each truck gets 10 mpg. During each month, the vehicles produced by the company must average at least 16 mpg. Formulate an LP to meet the demand and mileage requirements at minimum cost (include steel costs and holding costs).

Write the best name for each quadrilateral. A parallelogram with four congruent sides.

Consider the region bounded by $y = e^x,$ the x-axis, and the lines x = 0 and x = 1. Find the volume of the solid. The solid obtained by rotating the region about the horizontal line y = -3.

Find an equation of the circle with center $(-3,9)$ and radius 4

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Researchers from the University of British Columbia conducted a study to investigate the effects of color on cognitive tasks. Words were displayed on a computer screen with background colors of red and blue. Results from scores on a test of word recall are given below. Higher scores correspond to greater word recall.

$$\begin{array} { l } { \text { Red Background } \quad n = 35 , \overline { x } = 15.89 , s = 5.90 } \\ { \text { Blue Background } \quad n = 36 , \overline { x } = 12.31 , s = 5.48 } \end{array}$$

Does the background color appear to have an effect on word recall scores? If so, which color appears to be associated with higher word memory recall scores?

Divide.

$$14 ) \overline { 79.8 }$$

Solve the equation using the quadratic formula.

$$4 x ^ { 2 } - 11 x = 3$$