Let's break down each part of the problem, using the given function for the value of the MegaCash company:

The value $V$ of the company is modeled by the equation:

$V(t) = 2t^2 - 4t + 2$

Where:

• $V(t)$ is the value of the company (in millions of dollars),
• $t$ is the time in years, with $t = 0$ corresponding to the year 2000.

(a) What was the value of the company in 2002?

To find the value of the company in 2002, we substitute $t = 2$ (since $t = 0$ is the year 2000, so $t = 2$ corresponds to 2002):

$V(2) = 2(2)^2 - 4(2) + 2$

Let's calculate this:

$V(2) = 2(4) - 8 + 2 = 8 - 8 + 2 = 2$

So, the value of the company in 2002 was 2 million dollars.

(b) In what year(s) did the value of the company equal 1 million dollars?

To find when the value of the company was 1 million dollars, we set $V(t) = 1$ and solve for $t$:

$1 = 2t^2 - 4t + 2$

First, subtract 1 from both sides to set the equation equal to zero:

$0 = 2t^2 - 4t + 1$

Now, solve this quadratic equation. We can use the quadratic formula:

$t = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(1)}}{2(2)}$

$t = \frac{4 \pm \sqrt{16 - 8}}{4} = \frac{4 \pm \sqrt{8}}{4} = \frac{4 \pm 2\sqrt{2}}{4}$

$t = 1 \pm \frac{\sqrt{2}}{2}$

Approximating $\sqrt{2} \approx 1.414$, we get:

$t = 1 \pm \frac{1.414}{2} = 1 \pm 0.707$

Thus, the two possible values for $t$ are:

$t = 1 + 0.707 \approx 1.707 \quad \text{and} \quad t = 1 - 0.707 \approx 0.293$

These correspond to the years:

• $t = 1.707$ is approximately the year 2001.707, or late 2001.
• $t = 0.293$ is approximately the year 2000.293, or early 2000.

(c) Between what years did the company have to borrow money in order to survive?

The company would need to borrow money if its value drops below 0, meaning we need to find when $V(t) < 0$.

We set the equation $2t^2 - 4t + 2 = 0$ and solve for $t$:

$0 = 2t^2 - 4t + 2$

Dividing through by 2 to simplify:

$0 = t^2 - 2t + 1$

This is a perfect square trinomial:

$(t - 1)^2 = 0$

Thus, $t = 1$. This indicates that the value of the company equals 0 when $t = 1$, or the year 2001.

Since the quadratic function $2t^2 - 4t + 2$ opens upwards (the coefficient of $t^2$ is positive), the value of the company is positive for all $t \neq 1$. This suggests the company never went below zero; the function touches 0 at $t = 1$ but does not go negative. So, the company did not actually borrow money.

Summary of Answers:

• (a) The value of the company in 2002 was 2 million dollars.
• (b) The value of the company was 1 million dollars in early 2000 and late 2001.
• (c) The company never went below zero; hence, there was no need to borrow money.

Let me know if you'd like any further details or clarifications!

Follow-up Questions:

1. What would happen if the company had a different quadratic equation for its value?
2. How could we determine the maximum value of the company using calculus?
3. What factors could lead to the company needing to borrow money in the future?
4. How does the model compare to a linear model for company value?
5. What other financial metrics could be used to assess the company’s health?

Tip: When working with quadratic equations, always check the nature of the roots (whether they are real or complex) to understand the behavior of the function better!