# mathematics

Klein Industries manufactures three types of portable air compressors: small, medium, and large, which have unit profits of $20.50, $34.00, and $42.00, respectively. The projected monthly sales are:

Small | Medium | Large | |

Minimum | 14000 | 6200 | 2600 |

Maximum | 21000 | 12500 | 4200 |

The production process consists of three primary activities: bending and forming, welding, and painting. The amount of time in minutes needed to process each product in each department is shown below:

Small | Medium | Large | Available Time | |

Bending/forming | 0.4 | 0.7 | 0.8 | 23,400 |

Welding | 0.6 | 1 | 1.2 | 23,400 |

Painting | 1.4 | 2.6 | 3.1 | 46,800 |

How many of each type of air compressor should the company produce to maximize profit?

a. Formulate and solve a linear optimization model using the auxiliary variable cells method and write a short memo to the production manager explaining the sensitive information.

b. Solve the model without the auxiliary variables and explain the relationship between the reduced costs and the shodow prices found in part a.

The incomes of all families in a particular suburb can be represented by a continuous random variable. It is known that the median income for all families in this suburb is $60,000 and that 40% of all families in the suburb have incomes above $72,000.

a) For a randomly chosen family, what is the probability that its income will be between $60,000 and $72,000?

b) Given no further information is available, what can be said about the probability that a randomly chosen family has an income below $65,000?

A portfolio manager has asked you to analyze a newly acquired portfolio to determine its value and variability. The portfolio consists of 50 shares of Alpha Music and 40 shares of Beta Shows. Analysis of past history indicates that the share price of Alpha Music has a mean of $25 and variance of 121. Similarly, Beta Shows has a mean share price of $40 with a variance of 225. Your best evidence indicates that the share prices have a correlation coefficient of +0.5.

a) Compute the mean value and variance of the portfolio.

b) If the correlation coefficient between the share prices were actually – 0.5, what would be the mean value and the variance of the portfolio?

1. A manufacturer of network computer server systems is interested in improving the customer support service. As a first step, its marketing department has been charged with the responsibility of summarizing the extent of customer problems in terms of system downtime. The 40 most recent customers were surveyed to determine the amount of downtime in hours they had experienced in the previous month. The survey data are shown below.

a) Using the appropriate formula for grouped data, calculate the mean number of hours in downtime.

b) Using the appropriate formulas for grouped data, calculate the variance and the standard deviation.

c) Calculate the Coefficient of Variation