Question

The cost function for a certain company is C = 20x + 700 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 $700.

Answer 1

The two values of production level are:

x = 20

or

x = 140

Explanation:

Let's write the Profit equation by subtracting the cost function from the revenue one:

`Profit(x)=100\,x-0.5\,x^2-(20\,x+700)\\Profit(x)=-0.5\,x^2+80\,x-700`

Now, we set to find the production level "x" to give us a profit of $ 700:

`Profit(x)=-0.5\,x^2+80\,x-700\\700=-0.5\,x^2+80x-700\\0.5\,x^2-80\,x+1400=0`

So we solve this quadratic equation using the quadratic formula:

`x=(80+/-√(80^2-4(0.5)(7700)))/(2\,*\,0.5) \\x=80+/-√(3600) \\x=80\,+/-\,60`

which gives two posible solutions: x = 20, or x = 140