We have the cost function and the revenue function given as follows;
[tex]\begin{gathered} \text{Cost function:} \\ C=70x+540 \\ \text{ Revenue function:} \\ R=110x-0.4x^2 \end{gathered}[/tex]
The profit is calculated by subtracting cost from sales, that is;
[tex]\text{Profit}=\text{revenue}-\cos t[/tex]
Therefore, we would now have;
[tex]\begin{gathered} \text{Profit}=110x-0.4x^2-(70x+540) \\ \text{Profit}=110x-0.4x^2-70x-540 \\ \text{Profit}=110x-70x-0.4x^2-540 \\ \text{Profit}=40x-0.4x^2-540 \\ We\text{ can re-write this as;} \\ \text{Profit}=-0.4x^2+40x-540 \end{gathered}[/tex]
With the profit function we can now calculate the profit if we have the quantity produced (which is x). However, we have been given a weekly profit of $300, which means we can determine how many units are sold by inputing the profit into the profit function as follows;
[tex]\begin{gathered} \text{Profit}=-0.4x^2+40x-540 \\ 300=-0.4x^2+40x-540 \\ \text{Subtract 300 from both sides and you'll have;} \\ _{}-0.4x^2+40x-840=0 \end{gathered}[/tex]
We can now solve this quadratic equation as follows;
[tex]\begin{gathered} -0.4x^2+40x-840=0 \\ \text{Multiply all through by 10 to remove the decimal and we'll have;} \\ -4x^2+400-8400=0 \\ \text{Use the quadratic equation formula,} \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \text{Where,} \\ a=-4,b=400,c=-8400 \\ x=\frac{-400\pm\sqrt[]{(400)^2-4(-4)(-8400)}}{2(-4)} \\ x=\frac{-400\pm\sqrt[]{160000-134400}}{-8} \\ x=\frac{-400\pm\sqrt[]{25600}}{-8} \\ x=\frac{-400\pm160}{-8} \\ x=\frac{-400+160}{-8},x=\frac{-400-160}{-8} \\ x=30,x=70 \end{gathered}[/tex]
From the results, we can see that to make a profit of $300 per week, the company needs to sell 70 or 30 units of the product.
ANSWER:
They need to sell 70 or 30 items
