Answer : The level(s) of production will be, (60, 45)
Explanation :
As we are given the expression:
[tex]p(x)=105x-300-x^2[/tex]
The production yield a profit of $2400. That means,
[tex]p(x)=\$ 2400[/tex]
[tex]2400=105x-300-x^2[/tex]
Rearranging the terms, we get:
[tex]x^2-105x+300+2400=0[/tex]
[tex]x^2-105x+2700=0[/tex]
The general quadratic equation is,
[tex]ax^2+bx+c=0[/tex]
Formula used :
[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Now we have to solve the above equation and we get the value of 'x'.
[tex]x^2-105x+2700=0[/tex]
a = 1, b = -105, c = 2700
[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]x=\frac{-(-105)\pm \sqrt{(-105)^2-4\times 1\times (2700)}}{2\times 1}[/tex]
[tex]x=\frac{-(-105)+\sqrt{(-105)^2-4\times 1\times (2700)}}{2\times 1}[/tex]
x = 60
and,
[tex]x=\frac{-(-105)-\sqrt{(-105)^2-4\times 1\times (2700)}}{2\times 1}[/tex]
x = 45
The values of 'x' are 60 and 45.
Therefore, the level(s) of production will be, (60, 45)
