Question

The cost function for a certain company is c = 50x + 600 and the revenue is given by r = 100x − 0.5x2. 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 $600.

Answer 1

The quadratic equation is:  $p=-0.5x^2+50x-600$ and the two values of $x$ will be 40 and 60 that will create a profit of $600.

Explanation

The cost function is:  $c=50x+600$ and

The revenue function is:  $r=100x-0.5x^2$

As the profit means $(Revenue-Cost)$, so the Profit function will be....

$p= r-c\\ \\ p=(100x-0.5x^2)-(50x+600)\\ \\ p=100x-0.5x^2-50x-600\\ \\ p=-0.5x^2+50x-600$

So, the quadratic equation is:  $p=-0.5x^2+50x-600$

Now for creating a profit of $600 means, we will plug $p=600$ into the above equation. So.....

$600=-0.5x^2+50x-600\\ \\ 0.5x^2-50x+1200=0 \\ \\ 0.5(x^2-100x+2400)=0\\ \\ 0.5(x-40)(x-60)=0$

Now applying zero-product property.....

$x-40=0\\ x=40\\ \\ and\\ \\ x-60=0\\ x=60$

So, the two values of $x$ will be 40 and 60 that will create a profit of $600.