the total cost (in dollars) for a company to manufacture and sell $x$ items per week is $c = 70x + 1200$…

the total cost (in dollars) for a company to manufacture and sell $x$ items per week is $c = 70x + 1200$, whereas the revenue brought in by selling all $x$ items is $r = 118x - 0.4x^{2}$. how many items must be sold to obtain a weekly profit of $200? 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 $x$ items per week is $c = 70x + 1200$, whereas the revenue brought in by selling all $x$ items is $r = 118x - 0.4x^{2}$. how many items must be sold to obtain a weekly profit of $200? hint: profit = revenue - cost. they need to sell or items. question help: video ebook

Answer

Explanation:

Step1: Set up the profit equation

Profit $P=R - C$. Given $P = 200$, $R=118x - 0.4x^{2}$ and $C = 70x+1200$. So, $200=(118x - 0.4x^{2})-(70x + 1200)$.

Step2: Simplify the equation

$200=118x - 0.4x^{2}-70x - 1200$. Combine like - terms: $0=- 0.4x^{2}+(118x - 70x)-(1200 - 200)$. So, $0=-0.4x^{2}+48x - 1000$. Multiply through by 10 to clear the decimal: $0=-4x^{2}+480x - 10000$. Divide by - 4: $x^{2}-120x + 2500=0$.

Step3: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ (here $a = 1$, $b=-120$, $c = 2500$), use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(-120)^{2}-4\times1\times2500=14400 - 10000 = 4400$. Then $x=\frac{120\pm\sqrt{4400}}{2}=\frac{120\pm20\sqrt{11}}{2}=60\pm10\sqrt{11}$. $x_1=60 + 10\sqrt{11}\approx60+10\times3.317=60 + 33.17 = 93.17\approx93$ and $x_2=60 - 10\sqrt{11}\approx60-33.17 = 26.83\approx27$.

Answer:

27, 93