Question

The average cost for a product is C (x )equals 2 x plus 54 plus 98 over x, where C is measured in dollars and x is the number of units. How many units must be produced in order to minimize average cost

Answer 1

7 units

Explanation:

Given

`C(x) = 2x + 54 + (98)/(x)`

Required

The units that minimize the average cost

`C(x) = 2x + 54 + (98)/(x)`

Differentiate

`C' = 2 + 0 - 98x^(-2)`

`C' = 2 - 98x^(-2)`

Equate to 0 to solve for x

`0 = 2 - 98x^(-2)`

Collect like terms

`98x^(-2) = 2`

Divide by 98

`x^(-2) = (1)/(49)`

Rewrite as:

`(1)/(x^2) = (1)/(49)`

Take square roots of both sides

`(1)/(x) = \±(1)/(7)`

Take multiplicative inverse of both sides

`x = \±7`

Only positive value will produce critical value. Hence, 7 units will produce the minimum average cost