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

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!