The function that models the value $V$ of MegaCash company is given as:

$V(t) = 2.4t^2 - 2.02t + 2$

Where $V(t)$ is in millions of dollars, and $t$ is the number of years since 2000, i.e., $t = 0$ corresponds to the year 2000.

Let’s break down the questions:

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

To find the value in 2002, we need to substitute $t = 2$ (because 2002 is 2 years after 2000) into the equation.

$V(2) = 2.4(2)^2 - 2.02(2) + 2$ $V(2) = 2.4(4) - 2.02(2) + 2$ $V(2) = 9.6 - 4.04 + 2$ $V(2) = 7.56 \text{ million dollars}$

So, the value of the company in 2002 was 7.56 million dollars.

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(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, set $V(t) = 1$ and solve for $t$:

$1 = 2.4t^2 - 2.02t + 2$

Rearrange the equation:

$2.4t^2 - 2.02t + 1 = 0$

Now, solve this quadratic equation using the quadratic formula:

$t = \frac{-(-2.02) \pm \sqrt{(-2.02)^2 - 4(2.4)(1)}}{2(2.4)}$

$t = \frac{2.02 \pm \sqrt{4.0804 - 9.6}}{4.8}$ $t = \frac{2.02 \pm \sqrt{-5.5196}}{4.8}$

Since the discriminant is negative ($\sqrt{-5.5196}$), there are no real solutions, meaning the value of the company never equals 1 million dollars. Hence, there are no solutions for when $V(t) = 1$.

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(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, as it cannot operate at a loss.

To find when the value of the company was less than 0, we solve the inequality:

$2.4t^2 - 2.02t + 2 < 0$

First, solve the corresponding quadratic equation to find the critical points:

$2.4t^2 - 2.02t + 2 = 0$

Use the quadratic formula:

$t = \frac{-(-2.02) \pm \sqrt{(-2.02)^2 - 4(2.4)(2)}}{2(2.4)}$ $t = \frac{2.02 \pm \sqrt{4.0804 - 19.2}}{4.8}$ $t = \frac{2.02 \pm \sqrt{-15.1196}}{4.8}$

Again, the discriminant is negative, which means there are no real solutions. Since the quadratic does not cross the t-axis, the value of the company never goes below 0. Hence, the company never needed to borrow money to survive.

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Summary:

• (a) The value of the company in 2002 was 7.56 million dollars.
• (b) The value never equaled 1 million dollars.
• (c) The company never needed to borrow money to survive because its value never went below 0.

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

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Here are some related questions to deepen your understanding:

1. What happens to the value of the company as time progresses (large $t$)?
2. How does the quadratic function behave when the coefficient of $t^2$ changes?
3. What are the effects of changing the constant term in a quadratic function like this?
4. Can the function represent real-world situations in which growth is bounded (like market saturation)?
5. How do you interpret the negative discriminant in quadratic equations?

Tip: When solving quadratic equations with negative discriminants, it means there are no real solutions, and thus, the equation has no real roots (no x-intercepts).