 business that produces color copies is trying to minimize its average cost per copy​ (total cost divided by the number of​ copies). This average cost in cents is given by
​f(x)equals0.00000093x Superscript 2minus0.0138xplus60​, 
where x represents the total number of copies produced.
​(a) Describe the graph of f.
​(b) Find the minimum average cost per copy and the corresponding number of copies made.
Question content area bottom
Part 1
​(a) Describe the graph of f. Choose the correct answer below.
A.
a parabola that opens upward
Your answer is correct.B.
a parabola that opens downward
C.
a straight line
Part 2
​(b) The number of copies made at the minimum cost is 
  
enter your response here.

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.

---

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.

A business that produces color copies is trying to minimize its average cost per copy (total cost divided by the number of copies). This average cost in cents is given by f(x) = 0.00000093x^2 - 0.0138x + 60, where x represents the total number of copies produced. (a) Describe the graph of f. (b) Find the minimum average cost per copy and the corresponding number of copies made.

Learn how to minimize the average cost of producing color copies using a quadratic function. This problem involves graphing a parabola and finding the vertex, with key concepts including quadratic functions and optimization. The minimum cost per copy is approximately 51.93 cents when producing 7,419 copies.

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