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Question
A manufacturer can sell x items (x > 0) at a price of Rs.(280 - x) each. The cost of producing x items is Rs. (x2 + 40x + 35). Find the number of items to be sold so that the manufacturer can make maximum profit.
Solution
Selling price of an item = Rs. (280 - x)
Let x be the number of items to be sold
∴ Revenue = x( 280 - x )
= 280x - x2
Given total cost = Rs. (x2 + 40x + 35)
Profit π = Revenue - total cost
= (280x - x2) - (x2 + 40x + 35)
= -2x2 + 240x - 35
Diff. w.r.t.x.
∴ `(dπ)/dx = -2(2x) + 240 = 0`
= -4x + 240
π is maximum if, `(dπ)/dx` = 0
⇒ -4x + 240 = 0
⇒ -4x = -240
⇒ x = 60
Diff. again w.r.t.x.
`(d^2π)/dx^2 = (dpi)/dx (-4x + 240)`
= -4(1) + 0 = -4
`(d^2π)/dx^2 = -4 < 0`
Profit is maximum when x = 60
∴ Maximum profit when x = 60.
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