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I see that you've uploaded an image containing a business and mathematics problem related to Me Inc.'s "Total Life Tracker". Let's go through the questions in detail to assist you with the solutions.

### Part (a): Determine the Profit Function $ P(n) $

The profit $ P(n) $ is given by the difference between the revenue and the cost:
\[
P(n) = R(n) - C(n)
\]
We are provided with the cost function:
\[
C(n) = 350n + 3,500,000
\]
And the revenue function:
\[
R(n) = n \left( -\frac{1}{590000}n + 449 \right)
\]

Expanding the revenue function:
\[
R(n) = n \times \left( -\frac{1}{590000}n + 449 \right) = -\frac{n^2}{590000} + 449n
\]

Now, to find the profit function:
\[
P(n) = R(n) - C(n)
\]
\[
P(n) = \left( -\frac{n^2}{590000} + 449n \right) - (350n + 3,500,000)
\]
\[
P(n) = -\frac{n^2}{590000} + 449n - 350n - 3,500,000
\]
\[
P(n) = -\frac{n^2}{590000} + 99n - 3,500,000
\]

### Part (b): Vertical (y-) Intercept of $ P(n) $

The vertical intercept occurs when $ n = 0 $. Substituting $ n = 0 $ into the profit function:

\[
P(0) = -\frac{0^2}{590000} + 99(0) - 3,500,000 = -3,500,000
\]

So the vertical intercept is $ (0, -3,500,000) $.

#### Significance of the Vertical Intercept
The vertical intercept represents the initial cost or investment required for the production when no trackers are sold. In this case, it's the startup cost of $3,500,000, indicating that the company will start off with a significant expense.

### Part (c): Horizontal (n-) Intercepts of $ P(n) $

To find the horizontal intercepts, we set $ P(n) = 0 $:
\[
-\frac{n^2}{590000} + 99n - 3,500,000 = 0
\]
This is a quadratic equation. I will now solve this using the quadratic formula:
\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where:
- $ a = -\frac{1}{590000} $,
- $ b = 99 $,
- $ c = -3,500,000 $.

Let me calculate this.The horizontal intercepts (where $ P(n) = 0 $) occur at approximately $ n = 35,374.96 $ and $ n = 58,374,625.04 $.

#### Significance of the Horizontal Intercepts
These intercepts represent the number of Total Life Trackers that must be sold for the company to break even. The lower intercept (around 35,375) shows the minimum number of units that need to be sold to start making a profit, while the upper intercept (around 58.37 million) represents the point at which further production would not be profitable.

### Part (d): Vertex of $ R(n) $

The vertex of a quadratic function $ R(n) = -\frac{1}{590000}n^2 + 449n $ can be found using the formula:
\[
n_{\text{vertex}} = \frac{-b}{2a}
\]
Here, $ a = -\frac{1}{590000} $ and $ b = 449 $. Let's calculate the vertex next.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to proceed with any other calculations or clarifications? Let me know if you need help with anything else.

Here are 5 follow-up questions related to this topic:

1. How does the cost function impact the overall profit in this scenario?
2. What role does the quadratic nature of the revenue function play in determining maximum profit?
3. How can you interpret the significance of selling below the break-even point?
4. How would changes in production costs affect the intercepts?
5. Can the intercepts give insight into the scalability of the Total Life Tracker?

### Tip:
When solving for profit in business problems, always remember to check the constraints for production and revenue limits to ensure realistic solutions.

Me Inc. is creating a new fitness tracker that collects various data, such as heart rate, sleep quality, blood sugar levels, etc. The problem involves determining the profit function, vertical and horizontal intercepts, and more for the Total Life Tracker based on provided revenue and cost functions.

Explore the profit function, intercepts, and revenue maximization for Me Inc.'s Total Life Tracker. Solve for the quadratic profit function, identify intercepts, and determine the ideal price for maximizing profit. Perfect for Grades 10-12 math students.

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