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.

Answer 1

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.