Activity 14.
A company manufactures rope, and the total cost of producing \(r\) feet of rope is \(C(r)\) dollars.
(a)
What does it mean to say that \(C(2000)=800\text{?}\)
Answer.
It costs 800 dollars to produce 2000 feet of rope.
(b)
What are the units of \(C'(r)\text{?}\)
Answer.
dollars per foot
(c)
Suppose that \(C(2000)=800\) and \(C'(2000)=0.35\text{.}\) Estimate \(C(2100)\text{,}\) and justify your estimate by writing at least one sentence that explains your thinking.
Solution.
\(C(2100)\approx C(2000)+C'(2000)\cdot 100 = 800 + 0.35(100) = 835\) dollars. The cost of producing 2100 feet of rope is approximately the cost of producing 2000 feet. We add the marginal cost of producing rope above 2000 feet.
(d)
Do you think \(C'(2000)\) is less than, equal to, or greater than \(C'(3000)\text{?}\) Why?
Answer.
greater than; the cost to produce the 2001st foot of rope is likely more than the cost to produce the 3001st foot of rope due to economies of scale
(e)
Suppose someone claims that \(C'(5000)=-0.1\text{.}\) What would the practical meaning of this derivative value tell you about the approximate cost of the next foot of rope? Is this possible? Why or why not?
Hint.
If the approximate cost of producing the 5001st foot of rope is roughly \(-0.10\) dollars (i.e. one gets paid 10 cents to produce this foot), then perhaps an external (government?) subsidy is in play?
