Question

If the profit from the sale of x units of a product is P = 105x − 300 − x2, what level(s) of production will yield a profit of $2400? (Enter your answers as a comma-separated list.)

Answer 1

Answer : The level(s) of production will be, (60, 45)

Explanation :

As we are given the expression:

$p(x)=105x-300-x^2$

The production yield a profit of $2400. That means,

$p(x)=\ 2400$

$2400=105x-300-x^2$

Rearranging the terms, we get:

$x^2-105x+300+2400=0$

$x^2-105x+2700=0$

The general quadratic equation is,

$ax^2+bx+c=0$

Formula used :

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Now we have to solve the above equation and we get the value of 'x'.

$x^2-105x+2700=0$

a = 1, b = -105, c = 2700

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$x=\frac{-(-105)\pm \sqrt{(-105)^2-4\times 1\times (2700)}}{2\times 1}$

$x=\frac{-(-105)+\sqrt{(-105)^2-4\times 1\times (2700)}}{2\times 1}$

x = 60

and,

$x=\frac{-(-105)-\sqrt{(-105)^2-4\times 1\times (2700)}}{2\times 1}$

x = 45

The values of 'x' are 60 and 45.

Therefore, the level(s) of production will be, (60, 45)