If the total cost function for a product is C(x)=8(x+3)3C(x)=8(x+3)3 dollars, where x represents the number of hundreds of units produced, producing how many units will minimize the average cost? Find the minimum average cost

Given

• The expression C(x)=8(x+3)3C(x)=8(x+3)3

gives the total cost function, where xx represents the number of hundreds of units produced.

We have,

Average Cost is the total cost per unit of output. That is,

Average Cost =8(x+3)3x=8(x+3)3x .

When ddx(C(x)x)=0ddx(C(x)x)=0, the average cost is minimum. That is,

8(3(x+3)2)x−8(x+3)3×(1)x2=03x(x+3)2−(x+3)3=0(x+3)2(3x−x−3)=0(x+3)2(2x−3)=08(3(x+3)2)x−8(x+3)3×(1)x2=03x(x+3)2−(x+3)3=0(x+3)2(3x−x−3)=0(x+3)2(2x−3)=0

We have x≠−3x≠−3 because number of units can't be negative. Therefore, x=1.5x=1.5 is the correct option.

This gives, number of units that will minimize the average cost is 1.5×100=1501.5×100=150 units.

(b) Put x=1.5x=1.5 to calculate the minimum average cost dollars per hundred units. That is,

C(1.5)1.5=8(1.5+3)33C(1.5)1.5=243C(1.5)1.5=8(1.5+3)33C(1.5)1.5=243

Therefore, $ 243 243 is the minimum average cost per hundred dollars per hundred units.