Average Cost 1

[harpazo]

Suppose that the daily cost C of manufacturing bicycles is given by C(x) = 80x + 5000. Then the average daily cost is given by (x). How many bicycles must be produced each day for the average cost to be no more than $100?

Solution:

According to the textbook, the average cost = C(x)/x.

Why C(x)/x?

What is the definition of average cost (in words)?

100 = ( 80x + 5000)/x

Why is 100 equated to C(x)/x?

80 + (5000/x)

Calculate x.

20 = 5000/x

So, x = 250 should be the number of bicycles produced.

 

[MarkFL]

The average cost, just like any other arithmetic mean, will be the total divided by the number of data, which in this case is \(x\).

Instead of an equation, I would use a weak inequality:

[MATH]\frac{C(x)}{x}\le100[/MATH]
[MATH]\frac{80x+5000}{x}-100\le0[/MATH]
[MATH]\frac{250-x}{x}\le0[/MATH]
As it is tacitly assumed that \(0<x\), we thus find:

[MATH]250\le x[/MATH]
Thus, as long as 250 or more bicycles are produced, the average cost will not be greater than $100/bicycle.

 

[Dr.Peterson]

Suppose that the daily cost C of manufacturing bicycles is given by C(x) = 80x + 5000. Then the average daily cost is given by (x). How many bicycles must be produced each day for the average cost to be no more than $100?

I think you (or they) copied this carelessly, and it should be this:

Suppose that the cost C of manufacturing x bicycles is given by C(x) = 80x + 5000. Then the average cost of a bicycle is given by C(x)/x. How many bicycles must be produced each day for the average cost of a bicycle to be no more than $100?​

I'm not sure why the word "daily" is there; presumably it means they make x bicycles per day; but I would normally take "average daily cost" to mean the average total cost over some number of days. I don't have a high opinion of a source that would write a problem so poorly.

According to the textbook, the average cost = C(x)/x.

Why C(x)/x?

What is the definition of average cost (in words)?

The average cost of a bicycle is the total cost divided by the number of bicycles. The total cost of x bicycles is given as C(x), as I reconstructed the problem.

100 = ( 80x + 5000)/x

Why is 100 equated to C(x)/x?

Because if the average cost is $100, we can say that C(x)/x = 100.

As has been pointed out, an inequality is really more appropriate, since they didn't actually say that the average cost is $100, so what they are doing is looking for the boundary of the solution -- the number that puts you right on the edge.

Again, this source doesn't seem very reliable.

 

[harpazo]

Thank you everyone.