Question

Suppose the total cost, C(Q) of producing a quantity q of a product is given by the equation:

C(Q) = 5,000+5Q

Answer 1

The average cost per unit quantity A(q) = the total cost, C(q) divided by the quantity produced q. Find the limiting value of average cost per unit as q tends to infinity. in other words find

\[\lim_{q \rightarrow \infty} A(q)\]

Answer 2

Average cost A(q) = \[\lim_{q \rightarrow \infty }\frac{ 5,000+5q }{ q }\]

Answer 3

i'm wondering if the limit is 5

Answer 4

\[\frac{ 5,000 + 5q }{ q } = \frac{ 5000 }{ q } + \frac{ 5q}{ q } = \frac{ 5000 }{ q } + 5\]

Answer 5

My understanding is that as the quantity increases, or like as you increase the quantity by alot, the denominator explodes becoming much bigger than the numerator making the resulting fraction 0. so I think this would mean average cost = 0.

Answer 6

sorry I mean 0 for the fraction 5,000/q which would approach zero.

Answer 7

So q is growing infinite large. What you are looking for here is ceiling level of output for cost per unit q. To find this limit you take 1/q and multiply by top and bottom of this quotient \[\lim_{q \rightarrow \infty} = \frac{ (5000 + 5q) 1/q }{ q (1/q)}\]

\[\lim_{q \rightarrow \infty} = \frac{ \frac{ 5000 }{ q } + 5 }{ 1 }\]

Answer 8

so @snowsurf what does the result mean? like in practical terms for this problem?

Answer 9

so does this mean that our maximum average cost is 5?

Answer 10

So what it means as q get sufficiently large this is the lowest cost to produce this much product. 5 dollar per some quantity of q.

Answer 11

That's interesting, I got the whole evaluating the limit part.

i'm still trying to like interpret as to how it would be the lowest,

For whatever reason i thought this was a maximum, here's how I thought about it

so if Q the quantity gets bigger, 5,000 over Q gets smaller so I thought that 5 was the maximum,


\(\color{blue}{\text{Originally Posted by}}\) @snowsurf
So what it means as q get sufficiently large this is the lowest cost to produce this much product. 5 dollar per some quantity of q.
\(\color{blue}{\text{End of Quote}}\)

Answer 12

Oops I said that wrong. Most cost for the maximum output of q produced. So yes indeed it would be that maximum you would spend to produce some amount of product q.

Answer 13

thank!