Question

The total cost to produce a certain product is C(x)=6000+9x+0.2x where C(x) is in dollars and x is in items.

(a) How much does it cost in total to produce 100 items?

(b) Give an expression for the average cost function, C(x)=C(x)/x, including units.

(c) What is the average cost when producing 100 items, including units?

(d) Find the derivative of average cost, C'(x). Do you really need the quotient rule?

(e) What are the units on Cˉ (x) ?

Answer 1

The total cost to produce 100 items is $6920 and the average cost when producing 100 items is $69.20 per item. The derivative of the average cost function is 9.2 / x.

Step-by-step explanation:

(a) To find the total cost to produce 100 items, we plug in x = 100 into the cost function C(x)=6000+9x+0.2x:

C(100) = 6000 + 9(100) + 0.2(100) = 6000 + 900 + 20 = 6920

Therefore, it costs $6920 in total to produce 100 items.

(b) The average cost function is given by C(x) = C(x) / x, which represents the cost per item. In this case, C(x) = 6000 + 9x + 0.2x:

C(x) = (6000 + 9x + 0.2x) / x = (6000 + 9.2x) / x

(c) To find the average cost when producing 100 items, we substitute x = 100 into the average cost function:

C(100) = (6000 + 9.2(100)) / 100 = (6000 + 920) / 100 = 6920 / 100 = 69.20

Therefore, the average cost when producing 100 items is $69.20 per item.

(d) To find the derivative of the average cost function, C'(x), we differentiate the average cost function with respect to x:

C'(x) = (9.2x)' / x = 9.2 / x

Since C'(x) does not involve x^2 or higher powers of x, we do not need to use the quotient rule.

(e) The units on Cˉ (x) are dollars per item.