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प्रश्न
The total cost of manufacturing x articles C = 47x + 300x2 – x4 . Find x, for which average cost is decreasing
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उत्तर
Given, total cost function is
C = 47x + 300x2 – x4
Average cost CA = `"C"/"A"`
∴ CA = `(47 x + 300x^2 - x^4)/x`
= `(x(47 + 300x - x^3))/x`
∴ CA = 47 + 300x – x3
∴ `"dC"_"A"/"dx"` = 0 + 300x – 3x3
= 300x – 3x3
= 3(100 – x2)
Since average cost CA is a decreasing function `"dC"_"A"/"dx" < 0`
∴ 3(100 – x2) < 0
∴ 100 – x2 < 0
∴ 100 < x2
∴ x2 > 100
∴ x > 10 or x < – 10
But x < – 10 is not possible .....[∵ x > 0]
∴ x > 10
∴ The average cost CA is decreasing for x > 10.
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Differentiating w.r.t. x,
∴ `("dR")/("d"x) = square`
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∴ `("dR")/("d"x)` > 0
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Solution: Total cost C = 40 + 2x and Price p = 120 − x
Profit π = R – C
∴ π = `square`
Differentiating w.r.t. x,
`("d"pi)/("d"x)` = `square`
Since Profit is increasing,
`("d"pi)/("d"x)` > 0
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∴ x = 120 – p
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`("d"x)/("dp")` = `square`
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Ec = (0.0003)I2 + (0.075)I2
when I = 1000
