Question

The cost function for a certain company is C = 50x + 400 and the revenue is given by R = 100x − 0.5x^2. Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of x (production level) that will create a profit of $400.

Answer 1

The cost and revenue functions are given to be:

$\begin{gathered} C=50x+400 \\ R=100x-0.5x^2 \end{gathered}$

Recall that profit is revenue minus cost. If the profit is $400, we have that:

$R-C=400$

Therefore, we have:

$100x-0.5x^2-50x-400=400$

Rearranging, we have the equation to be:

$-0.5x^2+50x-800=0$

Solving the quadratic equation using the quadratic formula, we have:

$\begin{gathered} x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ a=-0.5,b=50,c=-800 \\ \therefore \\ x=\frac{-50\pm\sqrt{50^2-2(-0.5\times-800)}}{2\times-0.5} \end{gathered}$

Therefore, we can calculate the values of x to be:

$x=20,x=80$

Hence, the two values of x are 20 and 80.