Question

It costs 1212 dollars to manufacture and distribute a backpack. If the backpacks sell at x dollars​ each, the number​ sold, n, is given by n equals StartFraction 2 Over x minus 12 EndFraction plus 7 (100 minus x )n= 2 x−12+7(100−x). Find the selling price that will maximize profit.

Answer 1

The selling price that will maximize profit is $56.

Explanation:

Given : It costs 12 dollars to manufacture and distribute a backpack. If the backpacks sell at x dollars​ each, the number​ sold, n, is given by
`n=(2)/(x-12)+7(100-x)`

To find : The selling price that will maximize profit ?

Solution :

The cost price is $12.

The selling price is $x

Profit = SP-CP

Profit = x-12

The profit of n number is given by,

`P=(x-12)n`

Substitute the value of n,

`P=(x-12)((2)/(x-12)+7(100-x))`

`P=(2(x-12))/(x-12)+7(100-x)(x-12)`

`P=2+7(100x-1200-x^2+12x)`

`P=2+700x-8400-7x^2+84x`

`P=-7x^2+784x-8398`

Derivate w.r.t x,

`(dP)/(dx)=-14x+784`

Put it to zero for critical point,

`-14x+784=0`

`-14x=-784`

`x=(-784)/(-14)`

`x=56`

Derivate again w.r.t x, to determine maxima and minima,

`(d^2P)/(dx^2)=-14<0`

It is a maximum point.

Therefore, the selling price that will maximize profit is $56.