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.

Step-by-step 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.