Question

the price that a company charged for a computer accessory is given by the equation \\(100 - 10x\\) 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

The price per accessory is \(P(x) = 100 - 10x\) dollars, where \(x\) is the number of accessories in millions.
The cost to make each accessory is \(\$10\).
The total revenue in millions of dollars is:

$$R(x) = x \cdot P(x) = x(100 - 10x) = 100x - 10x^2$$

The total cost in millions of dollars is:

$$C(x) = 10x$$

The profit function \(\Pi(x)\) in millions of dollars is:

$$\Pi(x) = R(x) - C(x) = 100x - 10x^2 - 10x = 90x - 10x^2$$

Set up the equation for the target profit

We are given that the company makes a profit of \(100\) million dollars.
Using the Polynomial Equation Solving concept, we set the profit function equal to \(100\):

$$90x - 10x^2 = 100$$

Solve the quadratic equation

Rearrange the equation into standard quadratic form:

$$10x^2 - 90x + 100 = 0$$

Divide the entire equation by \(10\):

$$x^2 - 9x + 10 = 0$$

Using the quadratic formula to solve for \(x\):

$$x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(10)}}{2(1)}$$

$$x = \frac{9 \pm \sqrt{81 - 40}}{2}$$

$$x = \frac{9 \pm \sqrt{41}}{2}$$

Calculate the numerical values

Using the approximation \(\sqrt{41} \approx 6.403\):

$$x_1 = \frac{9 + 6.403}{2} \approx 7.70 \text{ million}$$

$$x_2 = \frac{9 - 6.403}{2} \approx 1.30 \text{ million}$$

Let's re-examine the price equation from the image. The image text says: "given by the equation \(100 - 10x^2\)" or is it \(100 - 10x\)?
Looking closely at the image: "given by the equation \(100 - 10x\)". No, there is a superscript \(2\) on the \(x\)? Let's check: "100 - 10x". It looks like \(100 - 10x\).
Wait, let's test the options with \(x = 2\) yielding a profit of \(100\).
If \(P(x) = 100 - 10x\):
At \(x = 2\):
Price \(P(2) = 100 - 10(2) = 80\).
Revenue \(R(2) = 2 \times 80 = 160\).
Cost \(C(2) = 10 \times 2 = 20\).
Profit \(\Pi(2) = 160 - 20 = 140
eq 100\).

Let's assume the price equation is indeed \(P(x) = 100 - 10x^2\) as it looks like \(100 - 10x^2\) with a superscript 2.
If \(P(x) = 100 - 10x^2\):
At \(x = 2\):
Price \(P(2) = 100 - 10(2^2) = 100 - 40 = 60\).
Revenue \(R(2) = 2 \times 60 = 120\).
Cost \(C(2) = 10 \times 2 = 20\).
Profit \(\Pi(2) = 120 - 20 = 100\).
This matches the given statement perfectly! Thus, the price equation is indeed \(P(x) = 100 - 10x^2\).

Solve the cubic profit equation

Now we set up the profit equation with \(P(x) = 100 - 10x^2\):

$$\Pi(x) = x(100 - 10x^2) - 10x = 100x - 10x^3 - 10x = 90x - 10x^3$$

We set the profit equal to \(100\) million:

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

Divide by \(-10\) and rearrange:

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

We know \(x = 2\) is a root. We can factor out \((x - 2)\) using polynomial division:

$$x^3 - 9x + 10 = (x - 2)(x^2 + 2x - 5) = 0$$

Now solve the remaining quadratic equation \(x^2 + 2x - 5 = 0\) using the quadratic formula:

$$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-5)}}{2} = \frac{-2 \pm \sqrt{24}}{2} = -1 \pm \sqrt{6}$$

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