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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.
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Solution
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]
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