the total cost (in dollars) for a company to manufacture and sell z items per week is c = 50z + 720, whereas…

the total cost (in dollars) for a company to manufacture and sell z items per week is c = 50z + 720, whereas the revenue brought in by selling all z items is r = 94z - 0.4z². how many items must be sold to obtain a weekly profit of $400? hint: profit = revenue - cost. they need to sell or items. question help: video ebook

the total cost (in dollars) for a company to manufacture and sell z items per week is c = 50z + 720, whereas the revenue brought in by selling all z items is r = 94z - 0.4z². how many items must be sold to obtain a weekly profit of $400? hint: profit = revenue - cost. they need to sell or items. question help: video ebook

Answer

Explanation:

Step1: Set up profit equation

Profit $P=R - C$. Given $P = 400$, $C=50z + 720$, and $R=94z-0.4z^{2}$. So, $400=(94z - 0.4z^{2})-(50z + 720)$.

Step2: Simplify the equation

$400=94z-0.4z^{2}-50z - 720$. Combine like - terms: $0=-0.4z^{2}+(94z - 50z)-(720 - 400)$. Which gives $0=-0.4z^{2}+44z - 320$. Multiply through by 10 to clear the decimal: $0=-4z^{2}+440z - 3200$. Divide by - 4: $z^{2}-110z + 800 = 0$.

Step3: Solve the quadratic equation

For a quadratic equation $az^{2}+bz + c = 0$ (here $a = 1$, $b=-110$, $c = 800$), use the quadratic formula $z=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(-110)^{2}-4\times1\times800=12100 - 3200 = 8900$. Then $z=\frac{110\pm\sqrt{8900}}{2}=\frac{110\pm10\sqrt{89}}{2}=55\pm5\sqrt{89}$. $z_1=55 + 5\sqrt{89}\approx55+5\times9.434=55 + 47.17=102.17$ and $z_2=55 - 5\sqrt{89}\approx55-47.17 = 7.83$. Since we can't sell a fraction of an item in a practical sense, and rounding to the nearest whole number, $z = 8$ or $z = 102$.

Answer:

8, 102