Question

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 million
3.45 million
40 million
48 million

Explanation:

Formulate the profit equation

Using the Polynomial Factoring knowledge point

$$
LATEXBLOCK0
$$

Set up the equation for the target profit

We are given that the profit is \(100\) million dollars:

$$ -10x^3 + 90x = 100 $$

Divide the entire equation by \(-10\):

$$ x^3 - 9x + 10 = 0 $$

Factor the cubic polynomial

We know \(x = 2\) is a given solution. We can use polynomial division or synthetic division to factor out \((x - 2)\):

$$ (x - 2)(x^2 + 2x - 5) = 0 $$

Solve the remaining quadratic equation

Find the roots of \(x^2 + 2x - 5 = 0\) using the quadratic formula:

$$
LATEXBLOCK1
$$

Since the number of accessories produced must be positive:

$$ x = -1 + \sqrt{6} \approx -1 + 2.449 = 1.45\text{ million} $$

Answer:

• (A) 1.45 million (Correct answer)
• (B) 3.45 million
• (C) 40 million
• (D) 48 million