Let us solve the problem step by step. The given function for profit is:

$P(x) = -0.5x^2 + 12x - 10$

Part (a): Find the year when the company reaches its maximum profit.

The function $P(x)$ is a quadratic equation, and since the coefficient of $x^2$ is negative ($-0.5$), the parabola opens downwards. The maximum value occurs at the vertex.

The formula to find the $x$-coordinate of the vertex for a quadratic function $ax^2 + bx + c$ is:

$x = -\frac{b}{2a}$

Here, $a = -0.5$ and $b = 12$. Substituting these values:

$x = -\frac{12}{2(-0.5)} = -\frac{12}{-1} = 12$

Thus, the company reaches its maximum profit after 12 years.

Part (b): What is the maximum profit the company can achieve?

To find the maximum profit, substitute $x = 12$ into the profit function:

$P(12) = -0.5(12)^2 + 12(12) - 10$

First, calculate each term:

$P(12) = -0.5(144) + 144 - 10$ $P(12) = -72 + 144 - 10$ $P(12) = 62$

The maximum profit is $62,000.

Part (c): In which years will the company’s profit be zero (break-even points)?

To find the break-even points, we need to solve $P(x) = 0$:

$-0.5x^2 + 12x - 10 = 0$

Divide through by $-0.5$ to simplify:

$x^2 - 24x + 20 = 0$

Now, solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 1$, $b = -24$, and $c = 20$. Substituting these values:

$x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4(1)(20)}}{2(1)}$ $x = \frac{24 \pm \sqrt{576 - 80}}{2}$ $x = \frac{24 \pm \sqrt{496}}{2}$ $x = \frac{24 \pm 22.27}{2}$

Now calculate the two solutions:

1. $x = \frac{24 + 22.27}{2} = \frac{46.27}{2} = 23.14$
2. $x = \frac{24 - 22.27}{2} = \frac{1.73}{2} = 0.87$

Thus, the company breaks even at approximately 0.87 years (about 10 months) and 23.14 years.

Final Answers:

(a) The company reaches its maximum profit after 12 years.
(b) The maximum profit is $62,000.
(c) The company’s profit is zero at approximately 0.87 years and 23.14 years.

Would you like detailed steps for any part of this solution?

Here are five related questions to consider:

1. How do you find the vertex for a general quadratic function?
2. What does it mean when the coefficient of $x^2$ is negative in a quadratic equation?
3. How can the discriminant ($b^2 - 4ac$) help determine the number of solutions to a quadratic equation?
4. What is the significance of break-even points in real-world business scenarios?
5. Can a quadratic model always be used to represent profit functions?

Tip: When solving quadratic equations, always simplify coefficients early to make calculations easier!