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

Answer:

7 units

Step-by-step explanation:

Given

$C(x) = 2x + 54 + \frac{98}{x}$

Required

The units that minimize the average cost

$C(x) = 2x + 54 + \frac{98}{x}$

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

$x^{-2} = \frac{1}{49}$

Rewrite as:

$\frac{1}{x^2} = \frac{1}{49}$

Take square roots of both sides

$\frac{1}{x} = \±\frac{1}{7}$

Take multiplicative inverse of both sides

$x = \±7$

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