Applied Maxima and Minima Problem

[jennifer361]

Homework Statement

Problem goes like this:
For a monopolist's product, the demand equation is: p=156-2q
_
and the average-cost is c(ave)=120+112/q

Homework Equations

We need to find the total cost function in terms of c=

The Attempt at a Solution

I can't seem to figure out how to change the average cost of cost...or maybe it's simple and I'm completely missing it...any help would be greatly appreciated!

 

[HallsofIvy]

I'm going to assume that p is the "demand"- the number of items purchased- at price c. (In the future it would be helpful to say things like that explicetely.) Then the revenue is pq= (156-2q)q= 156q- 2q². The average cost of each item 120- 112/q so the cost of q items is (120- 112/q)q= 120q- 112. (I have no idea what "average cost of cost" means!).

Finally, the net profit is the revenue minus the cost: 156q- 2q²- (120 q- 112)= 112+ 36q- 2q². You should be able to find the maximum value of that by completing the square.

 

[Architeuthis Dux]

I can understand your struggle with this problem. It seems like you are trying to find the total cost function in terms of c, which can be a bit challenging. However, let's break down the problem and see if we can find a solution.

First, let's define some terms. The demand equation, p=156-2q, represents the relationship between the price (p) and quantity (q) demanded by consumers. This equation is often referred to as the demand curve. On the other hand, the average-cost function, c(ave)=120+112/q, represents the relationship between the average cost (c) and quantity (q) produced by the monopolist.

Now, to find the total cost function, we need to understand that it is the sum of all costs incurred by the monopolist to produce a given quantity of the product. This includes both fixed costs (costs that do not change with the level of production) and variable costs (costs that vary with the level of production). In this case, the average-cost function represents the variable costs, while the fixed costs can be calculated by subtracting the variable costs from the total cost at a given quantity.

To find the total cost function in terms of c, we can use the following equation:

C(q) = c(ave) * q + F(q)

Where C(q) represents the total cost function, c(ave) represents the average-cost function, q represents the quantity produced, and F(q) represents the fixed costs.

In this case, we can substitute the average-cost function, c(ave)=120+112/q, into the equation and solve for F(q) to find the fixed costs:

C(q) = (120+112/q) * q + F(q)

C(q) = 120q + 112 + F(q)

F(q) = C(q) - 120q - 112

Now we have the fixed costs in terms of the total cost function. We can substitute this into the equation to find the total cost function in terms of c:

C(q) = c(ave) * q + (C(q) - 120q - 112)

C(q) = (120+112/q) * q + (C(q) - 120q - 112)

C(q) = 120q + 112 + C(q) - 120q - 112

C(q) = C