A car manufacturer produces 4 types of cars: compact, SUV, sedan, and custom, with sale prices set to $13,000, $28,000, $23,000 and $40,000 respectively. The manufacturingprocess involves two major steps: assembly and painting. The required numbers of hours pertype of car are as follows:

Compact SUV Sedan Custom
Assembly 4 9 7 10
Painting 1 1 3 40

The unumfacturer needs a production plait for one year in which there are 5,000 hours available for assembly and 4,250 man-hours for painting.
(a) Using xe = units of type i cars to produce, i = compact, SUV, sedan, and custom, formulate an LP (assuming this company can produce fractional number of cars) to determine an optimal manufacturing plan to maximize total revenue.
(b) write the dual of the LP in part a.

The monthly demand equation for an electric utility company is estimated to be p equals 60 minus left parenthesis 10 superscript negative 5 baseline right parenthesis x, where p is measured in dollars and x is measured in thousands of killowatt-hours. the utility has fixed costs of $3 comma 000 comma 000 per month and variable costs of $32 per 1000 kilowatt-hours of electricity generated, so the cost function is upper c left parenthesis x right parenthesis equals 3 times 10 superscript 6 baseline plus 32 x. (a) find the value of x and the corresponding price for 1000 kilowatt-hours that maximize the utility's profit. (b) suppose that the rising fuel costs increase the utility's variable costs from $32 to $38, so its new cost function is upper c 1 left parenthesis x right parenthesis equals 3 times 10 superscript 6 baseline plus 38 x. should the utility pass all this increase of $6 per thousand kilowatt-hours on to the consumers?