Let's address each part of the problem systematically:

Part 1: Describe the graph of $f(x) = 0.00000093x^2 - 0.0138x + 60$

The function is a quadratic equation in the form $ax^2 + bx + c$, where $a = 0.00000093$, $b = -0.0138$, and $c = 60$.

• Since the coefficient of $x^2$ (i.e., $a = 0.00000093$) is positive, the parabola opens upward.
• This means the function has a minimum point (vertex) rather than a maximum.

Thus, the correct answer is: A. A parabola that opens upward

Part 2: Find the minimum average cost per copy and the corresponding number of copies made

To find the minimum point of the quadratic function, we need to find the vertex of the parabola. The $x$-coordinate of the vertex for a quadratic function $ax^2 + bx + c$ is given by the formula:

$x = -\frac{b}{2a}$

Substitute $a = 0.00000093$ and $b = -0.0138$:

$x = -\frac{-0.0138}{2 \times 0.00000093} = \frac{0.0138}{0.00000186} = 7419.35$

Thus, the number of copies that minimize the average cost is approximately 7,419 copies.

Now, to find the minimum average cost, substitute $x = 7419.35$ into the original function:

$f(7419.35) = 0.00000093(7419.35)^2 - 0.0138(7419.35) + 60$

$f(7419.35) = 0.00000093(55047070.42) - 102.390 - 60$

$f(7419.35) \approx 51.93 \text{ cents}$

Thus, the minimum average cost per copy is approximately 51.93 cents when producing 7,419 copies.

Summary:

• (a) The graph is a parabola that opens upward.
• (b) The minimum average cost per copy is 51.93 cents, and it occurs when 7,419 copies are made.

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Do you have any questions, or would you like further details on any step?

Here are 5 related questions to explore further:

1. What would happen to the minimum cost if the coefficient of $x^2$ increased?
2. How would the cost function change if the linear term had a different sign?
3. What is the effect of producing more than the number of copies at minimum cost?
4. How can we interpret the $y$-intercept of the function $f(x)$?
5. How can we use the derivative to confirm the minimum point for this quadratic function?

Tip: The vertex formula $x = -\frac{b}{2a}$ is a quick way to find extrema (minimum/maximum points) in quadratic functions without completing the square.