Problems 1 through 4 (5 points each): Define labels for the variables in the fol

Question

Problems 1 through 4 (5 points each): Define labels for the variables in the following situations, and write the constraints and profit functions in algebraic form.

Examples:

Ex 1. A dealer has the goal of selling at least 200 vehicles (cars plus trucks) each month. Write a constraint on car and truck sales.

Ex 1 Answer: Let

cars = number of cars sold per month

trucks = number of trucks sold per month

Then

cars + trucks >= 200

Ex 2. A bakery earns $1.00 for selling a dozen donuts, $1.50 for selling a dozen croissants, and $0.75 for selling a dozen bagels. Write their profit function.

Ex 2 Answer: Let

p = profit

donut = dozens of donuts sold,

crois = dozens of croissants sold

bagel = dozens of bagels sold

Then

p = 1(donut) + 1.5(crois) +0.75(bagel).

__________________

Filing cabinet A has a footprint (floor space requirement) of 4 square ft. Cabinet B has a footprint of 6 square feet. The office only has 30 square feet that can be used for cabinets. Write a constraint on the number of cabinets that should be installed.

An aid agency is buying generators for storm survivors. The resettlement area requires at least 15,000 kWh. Generator A is rated at 150 kWh, generator B is rated at 175 kWh. Write a constraint prescribing the number of generators of each type that can be purchased.

Larry’s Yard Care mows lawns, trims hedges and winterizes flower beds. The company charges $15 per hour for mowing, $12 per hour for trimming, and $25 per hour for winterizing. Write an hourly income function for the company, in terms of the number of jobs, by type, that were underway during that hour.

Thunder Garage assembles diesel engines for cars and trucks. The car engines have four cylinders, the truck engines have six. Each cylinder in a diesel engine requires a separate fuel injector. The supplier can only provide 1300 injectors per month. Write a constraint prescribing the number of each type engine that can be produced.

(30%):

All-Round Bakery is a small specialty business in downtown Los Angeles that makes only two products: powdered sugar donuts and chocolate glazed donuts. Their largest customer is the LAPD, which has a daily standing order for 100 dozen sugar donuts and 100 dozen chocolate donuts. Owing to storage and supply constraints, All-Round can only stockpile enough ingredients for 400 doz sugar donuts and 200 doz chocolate donuts per day. All-Round’s specialty machinery can bake 10 dozen donuts in a batch. The time required for a batch of sugar donuts is 10 minutes, or 1 minute per dozen. The time required for a batch of chocolate donuts is longer, owing to lower temperature. A batch of chocolate donuts requires 20 minutes, or 2 minutes per dozen. The machinery can operate a maximum of 10 hours (600 minutes) per day.

All-Round makes a profit of $1.75 on a dozen sugar donuts, and $2.00 on a dozen chocolate glaze donuts. How many dozen of each type should they bake daily, to maximize their profit?

Explanation / Answer

1. Let the number of cabinet type A that can be installed be "a" and the number of cabinet type B that can be installed be "b".

Total footprint occupied by A = 4a and by B = 6b. Total footprint occupied by A and B = 4a+6b.

Constraint: 4a+6b<=30 (the total footpring occupied by A and B cannot exceed the available 30 square feet).

2. Let the number of A type generator be "a" and B type be "b". Power generated by A = number of generators*power of A = 150a. Power generated by B = number of generators*power of B = 175b

Total = 150a+175b.

Minimum requirement is 15,000kWh. Thus the constraint will be: 150a+175b>=15,000 (i.e. the minimum requirement should be met).

3. Let the number of jobs for mowing, trimming and winterizing be "a", "b" and "c" respectively. Hourly earning = sum of (no. of jobs*hourly rate)

= 15a+12b+25c. This is the houlry income function.

4. Let the number of engine types for car be "a" and trucks be "b". No. of cylinders for car = 4a, and for trucks = 6b (no. of engine*cylinder per engine)

Total = 4a+6b. Now, eacg cylinder requires a fuel injector. Thus number of fuel injectors = 4a+6b.

Constraint: 4a+6b<=1300 (requirement cannot exceed the availability).

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