Answer:
The selling price that will maximize profit is $56.
Step-by-step 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 [tex]n=\frac{2}{x-12}+7(100-x)[/tex]
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,
[tex]P=(x-12)n[/tex]
Substitute the value of n,
[tex]P=(x-12)(\frac{2}{x-12}+7(100-x))[/tex]
[tex]P=\frac{2(x-12)}{x-12}+7(100-x)(x-12)[/tex]
[tex]P=2+7(100x-1200-x^2+12x)[/tex]
[tex]P=2+700x-8400-7x^2+84x[/tex]
[tex]P=-7x^2+784x-8398[/tex]
Derivate w.r.t x,
[tex]\frac{dP}{dx}=-14x+784[/tex]
Put it to zero for critical point,
[tex]-14x+784=0[/tex]
[tex]-14x=-784[/tex]
[tex]x=\frac{-784}{-14}[/tex]
[tex]x=56[/tex]
Derivate again w.r.t x, to determine maxima and minima,
[tex]\frac{d^2P}{dx^2}=-14<0[/tex]
It is a maximum point.
Therefore, the selling price that will maximize profit is $56.
