The total cost (in hundreds of dollars) to produce x units of a product is C(x)=8x−93x+5C(x)=8x−93x+5

a. To find the average cost of producing 15 items, we can calculate the total cost of producing 15 units first.

C(15)=8(15)−93(15)+5=11150=2.22C(15)=8(15)−93(15)+5=11150=2.22

The total cost of producing 15 items is thus $2.22 hundred, or $222. We can now find the average cost per item by dividing this by the 15 items produced.

¯C(15)=22215≈14.8C¯(15)=22215≈14.8

Thus, each item costs approximately $14.80 to produce.

b. Let's now find a general function that gives this average cost. This can be found by dividing the function by x.

¯C(x)=C(x)x=8x−93x+5x=8x−9x(3x+5)=8x−93x2+5xC¯(x)=C(x)x=8x−93x+5x=8x−9x(3x+5)=8x−93x2+5x

c. Now that we have an average cost function, we can use it to find the marginal average cost function. This is the same as the derivative of the average cost function. Since this average cost function is defined as a quotient, we have to use the quotient rule. In order to use the quotient rule, we need to define four pieces of information.

f=8x−9f′=8g=3x2+5xg′=6x+5f=8x−9f′=8g=3x2+5xg′=6x+5

Let's now combine this information together to find the marginal average cost function.

¯C′(x)=f′g−fg′g2=8(3x2+5x)−(8x−9)(6x+5)(3x2+5x)2=24x2+40x−(48x2−54x+40x−45)(3x2+5x)2=−24x2+54x+45(3x2+5x)2