$$ C = - 6.8 x ^ { 2 } + 7795 x + 230,000 ( x \leq 150 ) $$

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Certainly! Let's break down and analyze the given equation:

$$ C = -6.8x^{2} + 7,\!795x + 230,\!000 \quad \text{for} \quad x \leq 150 $$

**Understanding the Equation:**

1. **Variables and Constants:**
   - $$ C $$: This typically represents the total cost, revenue, or another quantity dependent on $$ x $$.
   - $$ x $$: This is the independent variable, which could represent units produced, hours worked, or another measurable factor.
   - The coefficients and constants:
     - $$ -6.8x^{2} $$: A quadratic term indicating that the relationship between $$ C $$ and $$ x $$ is not linear. The negative coefficient suggests that after a certain point, increases in $$ x $$ will lead to a decrease in $$ C $$.
     - $$ 7,\!795x $$: A linear term contributing positively to $$ C $$.
     - $$ 230,\!000 $$: A constant term representing the base value of $$ C $$ when $$ x = 0 $$.

2. **Domain Restriction:**
   - $$ x \leq 150 $$: The equation is valid for values of $$ x $$ up to and including 150. This could represent a production limit, availability of resources, or another constraint.

**Analyzing the Equation:**

1. **Nature of the Function:**
   - The equation is a quadratic function of the form $$ C = ax^{2} + bx + c $$, where:
     - $$ a = -6.8 $$
     - $$ b = 7,\!795 $$
     - $$ c = 230,\!000 $$
   - Since $$ a < 0 $$, the parabola opens downward, indicating that the function has a maximum point.

2. **Finding the Vertex (Maximum Point):**
   - The vertex of a parabola given by $$ ax^{2} + bx + c $$ occurs at $$ x = -\frac{b}{2a} $$.
   - Plugging in the values:
     $$
     x = -\frac{7,\!795}{2 \times (-6.8)} = \frac{7,\!795}{13.6} \approx 572.01
     $$
   - However, since the domain is restricted to $$ x \leq 150 $$, the maximum within this domain occurs at $$ x = 150 $$.

3. **Calculating $$ C $$ at $$ x = 150 $$:**
   $$
   C = -6.8 \times (150)^{2} + 7,\!795 \times 150 + 230,\!000
   $$
   $$
   C = -6.8 \times 22,\!500 + 1,\!169,\!250 + 230,\!000
   $$
   $$
   C = -153,\!000 + 1,\!169,\!250 + 230,\!000
   $$
   $$
   C = 1,\!246,\!250
   $$

4. **Behavior of $$ C $$ within the Domain:**
   - As $$ x $$ increases from 0 to 150:
     - The $$ -6.8x^{2} $$ term initially causes $$ C $$ to decrease.
     - The $$ 7,\!795x $$ term causes $$ C $$ to increase at a diminishing rate.
     - The combined effect reaches its peak at the boundary $$ x = 150 $$, as calculated above.

5. **Graphical Representation:**
   - The graph of $$ C $$ vs. $$ x $$ would be a downward-opening parabola.
   - Within the domain $$ x \leq 150 $$, the function increases up to $$ x = 150 $$.

**Possible Applications:**

- **Revenue Analysis:** If $$ C $$ represents revenue, the company maximizes revenue at $$ x = 150 $$ within the given constraints.
- **Cost Minimization:** If $$ C $$ represents cost, the company would aim to minimize cost at a certain production level, but given the structure of the equation, costs decrease as $$ x $$ increases up to 150.

**Conclusion:**

The equation $$ C = -6.8x^{2} + 7,\!795x + 230,\!000 $$ models a scenario where $$ C $$ changes with $$ x $$ up to a maximum $$ x $$ value of 150. The maximum value of $$ C $$ within this domain occurs at $$ x = 150 $$, resulting in $$ C = 1,\!246,\!250 $$. Understanding the context (whether $$ C $$ represents cost, revenue, or another metric) is crucial for making informed decisions based on this equation.

If you have a specific question or need further analysis on this equation, feel free to ask!
The equation $$ C = -6.8x^{2} + 7,\!795x + 230,\!000 $$ models how $$ C $$ changes with $$ x $$ up to $$ x = 150 $$. The maximum value of $$ C $$ is 1,\!246,\!250 when $$ x = 150 $$.

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