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?\n1.45 million\n3.45 million\n40 million\n48 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?\n1.45 million\n3.45 million\n40 million\n48 million

Answer

Explanation:

Step1: Define profit formula

Profit $P=(100 - 10x^{2})x-10x$.

Step2: Substitute known - value

Since $P = 100$ when $x = 2$, we have the equation $(100 - 10x^{2})x-10x=100$. Expand it: $100x-10x^{3}-10x = 100$, which simplifies to $- 10x^{3}+90x - 100 = 0$. Divide through by $-10$ to get $x^{3}-9x + 10 = 0$.

Step3: Factor the equation

We know that $x = 2$ is a root. So we can factor $x^{3}-9x + 10$ as $(x - 2)(x^{2}+2x - 5)=0$.

Step4: Solve the quadratic equation

For the quadratic equation $x^{2}+2x - 5=0$, use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a = 1$, $b = 2$, and $c=-5$. Then $x=\frac{-2\pm\sqrt{4+20}}{2}=\frac{-2\pm\sqrt{24}}{2}=\frac{-2\pm2\sqrt{6}}{2}=-1\pm\sqrt{6}$. We take the positive root $x=-1+\sqrt{6}\approx - 1 + 2.45=1.45$ (since the number of accessories $x>0$).

Answer:

A. 1.45 million