Part 1: Form the Profit Function $P(x)$

To create the profit function, we subtract the cost function $C(x)$ from the revenue function $R(x)$.

Given:

• Revenue function $R(x) = x(70 - 5x) = 70x - 5x^2$
• Cost function $C(x) = 121 + 13x$

The profit function $P(x)$ is:

$P(x) = R(x) - C(x) = (70x - 5x^2) - (121 + 13x)$ Simplifying $P(x)$:

$P(x) = 70x - 5x^2 - 121 - 13x$ $P(x) = -5x^2 + 57x - 121$

Thus, the profit function is:

$P(x) = -5x^2 + 57x - 121$

Part 2: Which Graph Represents $R(x)$, $C(x)$, and $P(x)$?

We are provided with graphs and need to match the functions to their visual representations based on their characteristics.

• $R(x) = 70x - 5x^2$ is a downward-facing parabola with a vertex at $x = \frac{-70}{2(-5)} = 7$, and it opens downwards since the coefficient of $x^2$ is negative. Its $x$-intercepts (where $R(x) = 0$) occur when $70x - 5x^2 = 0$, or $x(70 - 5x) = 0$, giving $x = 0$ and $x = 14$.

• $C(x) = 121 + 13x$ is a straight line with a slope of 13 and a y-intercept of 121.

• $P(x) = -5x^2 + 57x - 121$ is another downward-facing parabola, with its vertex and intercepts derived from the equation above.

Key Features to Match with the Graphs:

• The dashed curve $R(x)$ should be a downward-facing parabola with intercepts at $x = 0$ and $x = 14$, and a vertex at $x = 7$, with a maximum around 250.
• The line $C(x)$ starts at 121 and increases steadily with slope 13.
• The solid curve $P(x)$ should be a downward-facing parabola with $x$-intercepts at around $x = 2.8$ and $x = 8.6$, and a vertex lower than that of $R(x)$.

Conclusion:

Based on these characteristics, the correct graph is option B:

• The dashed parabola $R(x)$ has intercepts at $x = 0$ and $x = 14$, with a vertex at approximately $(7, 250)$.
• The solid parabola $P(x)$ has intercepts at around $x = 2.8$ and $x = 8.6$, with a vertex at around $(5.7, 40)$.
• The line $C(x)$ rises with a y-intercept of 121.

Would you like more details or have any other questions?

Additional Related Questions:

1. How do you calculate the break-even points using the profit function?
2. What is the maximum profit the company can achieve, and at what production level does it occur?
3. How do you determine the range where the company is making a profit (i.e., $P(x) > 0$)?
4. What does the vertex of the revenue function $R(x)$ represent in terms of production?
5. How would the graph change if the cost of manufacturing chips increased by $2 per million?

Tip:

The vertex of a parabola in a revenue or profit function often indicates the optimal production level to maximize revenue or profit.