Answer : The level(s) of production will be, (60, 45)
Explanation :
As we are given the expression:
![p(x)=105x-300-x^2](https://tex.z-dn.net/?f=p%28x%29%3D105x-300-x%5E2)
The production yield a profit of $2400. That means,
![p(x)=\$ 2400](https://tex.z-dn.net/?f=p%28x%29%3D%5C%24%202400)
![2400=105x-300-x^2](https://tex.z-dn.net/?f=2400%3D105x-300-x%5E2)
Rearranging the terms, we get:
![x^2-105x+300+2400=0](https://tex.z-dn.net/?f=x%5E2-105x%2B300%2B2400%3D0)
![x^2-105x+2700=0](https://tex.z-dn.net/?f=x%5E2-105x%2B2700%3D0)
The general quadratic equation is,
![ax^2+bx+c=0](https://tex.z-dn.net/?f=ax%5E2%2Bbx%2Bc%3D0)
Formula used :
![x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-b%5Cpm%20%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
Now we have to solve the above equation and we get the value of 'x'.
![x^2-105x+2700=0](https://tex.z-dn.net/?f=x%5E2-105x%2B2700%3D0)
a = 1, b = -105, c = 2700
![x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-b%5Cpm%20%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
![x=\frac{-(-105)\pm \sqrt{(-105)^2-4\times 1\times (2700)}}{2\times 1}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-%28-105%29%5Cpm%20%5Csqrt%7B%28-105%29%5E2-4%5Ctimes%201%5Ctimes%20%282700%29%7D%7D%7B2%5Ctimes%201%7D)
![x=\frac{-(-105)+\sqrt{(-105)^2-4\times 1\times (2700)}}{2\times 1}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-%28-105%29%2B%5Csqrt%7B%28-105%29%5E2-4%5Ctimes%201%5Ctimes%20%282700%29%7D%7D%7B2%5Ctimes%201%7D)
x = 60
and,
![x=\frac{-(-105)-\sqrt{(-105)^2-4\times 1\times (2700)}}{2\times 1}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-%28-105%29-%5Csqrt%7B%28-105%29%5E2-4%5Ctimes%201%5Ctimes%20%282700%29%7D%7D%7B2%5Ctimes%201%7D)
x = 45
The values of 'x' are 60 and 45.
Therefore, the level(s) of production will be, (60, 45)