Algebra Problem

Question

A company that manufactures wheelchairs has a fixed costs of$500,000. The average cost per wheel chair, C, for the company to manufacture x wheelchairs per month is modeled by the formula.

C=700x+500,000/x.

Answer 1

Not sure exactly what the question is, although here are a few things you might be looking for:

C=700+500,000x

Often the variable is AC for Average Cost

AC=700+500,000/x

Then TC = total cost

TC=700x+500,000 = Variable Cost + Fixed Cost

Take the derivative to get marginal cost

MC=700

set MC=MR to get maximum profits

Marginal cost = Marginal Revenue

In the SR, Short run, you keep producing as long as you can cover AC, average costs

In the LR, you may shut down if you are no longer profitable.

The breakeven point is where TC=TR total cost = total revenue

TC=700x+500,000 =PQ= price of each wheel chair times the quantity sold =Px

You breakeven when 700x+500,000=Px

Px-700x=500,000

x(P-700)=500,000

There's a revenue function that relates P to x. The higher P, the lower sales. Reduce price and sales rise.

To cover AC, you need a price of at least 700.

To cover TC, you need a price that depends on the revenue function. Plot price on the vertical axis and quantity sold on the horizontal axis. It's a downward sloping demand curve

Let's say it's something like x=10,000-P Drop price to zero and you can only give away 10,000 wheelchairs.

Raise the price to 700 and you can only sell 9300. Raise the price to 10,000 and no one buys anything.

substitute into x(P-700)=500,000

to get (10000-P)(P-700)=500,000

10000P+700P-700,000-P²

P²+10,0700P-700,000=0

solve for P, use completing the square

P²-10,0700P+(5,0350)² -700,000-(5,0350)²=0

(P-5,0350)^1/2=[700,000+(5,0350)²]^1/2

....etc.

To maximize profits find MC and MR and set equal

MC = 700, MR =2P-10,500 the first derivative of the total revenue function

700=2P-10,500

2P=11,200

P=5,600

quantity sold = x = 10,000-5,600 = 4,400

TR=xP = 4,400(5,600) = 24,640,000

TC=x700+500,000 = 4,400(700)+500,000=3,080,000+500,000=3,580,000

profits = TR-TC=24,640,000-3,580,000=$21,060,000

maximum profits are 21.06 million when price = $5,600 selling 4,400 wheel chairs.