Let's analyze the problem.

The function provided is: $f(x) = 0.00000093x^2 - 0.0141x + 65$ where $x$ represents the total number of copies produced, and $f(x)$ represents the average cost per copy in cents.

Part (a): Describe the graph of $f$

The function $f(x)$ is a quadratic equation in the form $ax^2 + bx + c$, where:

• $a = 0.00000093$ (positive),
• $b = -0.0141$, and
• $c = 65$.

Since the coefficient of $x^2$ (i.e., $a$) is positive, this quadratic function opens upward. So the answer is:

• A. A parabola that opens upward.

Part (b): Find the minimum average cost per copy and the corresponding number of copies

To find the minimum point of a parabola that opens upward, we use the vertex formula: $x = -\frac{b}{2a}$ Substituting the values of $a$ and $b$: $x = -\frac{-0.0141}{2 \times 0.00000093}$ $x = \frac{0.0141}{0.00000186}$ $x \approx 7581$ So, the number of copies that minimizes the average cost per copy is approximately 7,581 (rounded to the nearest whole number).

Calculating the minimum average cost

To find the minimum average cost, substitute $x = 7581$ back into the function $f(x)$: $f(7581) = 0.00000093 \times (7581)^2 - 0.0141 \times 7581 + 65$ Calculating each part:

1. $0.00000093 \times (7581)^2 \approx 53.56$
2. $-0.0141 \times 7581 \approx -106.89$
3. $65$

Adding these results: $f(7581) \approx 53.56 - 106.89 + 65 \approx 11.67$ So, the minimum average cost per copy is approximately 11.67 cents.

Summary

• (a) The graph of $f$ is a parabola that opens upward.
• (b) The minimum average cost per copy is approximately 11.67 cents, achieved at around 7,581 copies.

Would you like more details on any part of the solution, or have any questions?

Related Questions

1. How does the value of $a$ in a quadratic function determine the direction of the parabola?
2. What is the importance of finding the vertex in optimization problems involving parabolas?
3. How do you interpret the minimum cost in a real-world business context?
4. Can there be multiple minimum points in a quadratic function?
5. How does rounding affect the accuracy of optimization problems in practical applications?

Tip

When solving optimization problems, remember that if the quadratic term's coefficient is positive, the function opens upward, indicating a minimum point at the vertex.