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प्रश्न
A monopolist has a demand curve x = 106 – 2p and average cost curve AC = 5 + `x/50`, where p is the price per unit output and x is the number of units of output. If the total revenue is R = px, determine the most profitable output and the maximum profit.
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उत्तर
x = 106 – 2p
(or) 2p = 106 – x
p = `1/2`(106 – x)
Revenue, R = px
= `1/2`(106 – x) x
= 53x – `x^2/2`
Average Cost, AC = `5 + x/50`
Cost C = (AC)x
= `(5 + x/50)x`
= `5x + x^2/50`
Profit (P) = Revenue – Cost
`"dP"/"dx" = 48 - (13(2x))/25`
`"dP"/"dx"` = 0 gives
`48 - (13(2x))/25` = 0
`48 = (13 xx 2x)/25`
x = `(48 xx 25)/(13 xx 2)`= 46.1538 = 46 (approximately)
Also `("d"^2"P")/"dx"^2 = 0 - (13)^2/25`, negative since `("d"^2"P")/"dx"^2` is negative, profit is maximum at x = 46 units.
Profit = `48x – 13/25` x2
When x = 46,
Profit = `48 × 46 - 13/25` × 46 × 46
`= 2208 - 27508/25`
= 2208 – 1100.32
= ₹ 1107.68
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