Course:MATH110/Archive/2010-2011/003/Teams/Basel/homework11

Group Project - Homework 11

Due January 19,2011

Problem 3

According to the problem, the marginal cost* of 1 single Basel flag is $7. If 20 flags are produced, the price per flag goes down to $5.

The first question that arises, is what would be an appropriate model to show what it costs for x amount of flags produced.

Model

f = number of flags produced

P = total price

P(f) = 7(f) ; when f < 20

• This model represents the cost of the flags when the quantity purchased is less than 20 units.
• As we stated above, it costs $$100 to produce 20 flags (at $5/flag) in comparison to if the flags were $7 each, it would cost $140. This gives us a difference of $40.
• To account for the $40 in savings from $7/flag to $5/flag, we input the -40 into our original model.

P(f)=7(f)-40 ; when f ≥ 20

Finally, find some other models (not necessarily linear) for which you get other behaviours such as:

   ◦    The average cost remains constant as production increases:

f(x) = 10x

Where the cost, f, is linear in relation to x, the number of items and it costs $10 to produce x amount of items. This model is one in which the average cost of producing each item remains the same even as the rate of production increases.

   ◦    The average cost diminishes as production increases.

f(x) = 1000/x

Where cost, f, decreases at a constant rate as x increases. As x approaches infinity, the average cost of producing each item decreases. The less items you produce, the more it will cost.

   ◦ The average cost increases as production increases.

f(x) = x^2

Where cost, f, increases at a constant rate as x increases. The cost to produce x amount of items is the number of items to the power of 2. As x approaches infinite, the average cost of producing each item increases. The more items produced, the more it will cost.

   ◦    You obtain an economy of scale. This means that starting at some specific production level, the marginal cost is always less than the average cost.

f(x) = 5x + 2  ; x < 500

f(x) = 4 ; x ≥ 500

Where cost to produce 1 item is 5$ plus 2 dollars production charge per each batch of items produced. After you have produced 500 items, the cost to produce any number of additional items in that batch will only cost an additional $4.

   ◦    Any other interesting properties that you can think of and create a model for. Bonus points can be obtained for very interesting ideas.

f(x) = (x-5)^2 + 10

Where cost, f, increases to a peak cost of $10 when 5 items are produced. Any number of items produced less than or more than 5 will cost less than $10.

• *Marginal cost - the addition to the total cost when the quantity of the product produced increases by one unit.