Question

The cost of producing x articles is given by C = x² + 15x + 81. Find the average cost and marginal cost functions. Find marginal cost when x = 10. Find x for which the marginal cost equals the average cost.

Answer 1

Given, cost C = x² + 15x + 81

Average cost = `"C"/x=(x^2+15x+81)/x`

= x + 15 + `81/x`

and Marginal cost = `("dC")/("d"x)`

= `"d"/("d"x)(x^2 + 15x + 81)`

= `"d"/("d"x)(x^2) + 15d/("d"x)(x) + "d"/("d"x)(81)`

= 2x + 15(1) + 0
= 2x + 15
When x = 10,

Marginal cost = `(("dC")/("d"x))_(x = 10)`

= 2(10) + 15
= 35
If marginal cost = average cost, then

2x + 15 = x + 15 + `81/x`

∴ x = `81/x`

∴ x² = 81
∴ x = 9   …[∵ x > 0]