the price that a company charged for a computer accessory is given by the equation $100-10x^{2}$ where x is…

the price that a company charged for a computer accessory is given by the equation $100-10x^{2}$ where x is the number of accessories that are produced, in millions. it costs the company $10 to make each accessory. the company currently produces 2 million accessories and makes a profit of 100 million dollars. what other number of accessories produced yields the same profit?1.45 million3.45 million40 million48 million

the price that a company charged for a computer accessory is given by the equation $100-10x^{2}$ where x is the number of accessories that are produced, in millions. it costs the company $10 to make each accessory. the company currently produces 2 million accessories and makes a profit of 100 million dollars. what other number of accessories produced yields the same profit?1.45 million3.45 million40 million48 million

Answer

Explanation:

Step1: Define profit formula

Profit = Revenue - Cost. Revenue = Price × Quantity = $(100-10x^2)x$, Cost = $10x$. So Profit $P = x(100-10x^2) - 10x$.

Step2: Simplify profit equation

$P = 100x - 10x^3 - 10x = 90x - 10x^3$. Set $P=100$: $$90x - 10x^3 = 100$$ Rearrange to: $10x^3 - 90x + 100 = 0$, divide by 10: $x^3 - 9x + 10 = 0$.

Step3: Factor known root

We know $x=2$ is a root. Use polynomial division or factor theorem: $(x-2)(x^2 + 2x - 5) = 0$

Step4: Solve quadratic equation

For $x^2 + 2x - 5 = 0$, use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1,b=2,c=-5$: $$x=\frac{-2\pm\sqrt{4 + 20}}{2}=\frac{-2\pm\sqrt{24}}{2}=-1\pm\sqrt{6}$$

Step5: Select positive root

$\sqrt{6}\approx2.45$, so positive root: $x=-1+2.45=1.45$ (discard negative root $-1-2.45=-3.45$ as quantity can't be negative)

Answer:

1.45 million