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)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.
​(Round to the nearest whole number as​ needed.)

Let's solve each part of this problem step-by-step.

### Part (a) Describe the Graph of \( f(x) \)

Given the function:
\[
f(x) = 0.00000093x^2 - 0.0138x + 60
\]
The function \( f(x) \) is a quadratic equation in terms of \( x \), and its graph will be a parabola. To determine the direction in which the parabola opens, we observe the leading coefficient of \( x^2 \), which is \( 0.00000093 \). Since this coefficient is positive, the parabola opens **upward**.

**Answer for Part (a):** The correct answer is **A. A parabola that opens upward.**

### Part (b) Finding the Minimum Average Cost and the Corresponding Number of Copies

To find the minimum point of the parabola, we calculate the vertex of the quadratic function. For a quadratic function \( ax^2 + bx + c \), the \( x \)-coordinate of the vertex (where the minimum or maximum occurs) is given by:
\[
x = -\frac{b}{2a}
\]
For our function, \( a = 0.00000093 \) and \( b = -0.0138 \).

1. Calculate \( x \) at the vertex:
   \[
   x = -\frac{-0.0138}{2 \times 0.00000093} = \frac{0.0138}{0.00000186} \approx 7419.35
   \]
   Rounding to the nearest whole number, we get:
   \[
   x \approx 7419
   \]

2. Substitute \( x = 7419 \) into the function \( f(x) \) to find the minimum average cost:
   \[
   f(7419) = 0.00000093(7419)^2 - 0.0138(7419) + 60
   \]
   Calculating each term:
   \[
   f(7419) \approx 0.00000093 \times 55041261 - 0.0138 \times 7419 + 60
   \]
   \[
   f(7419) \approx 51.88 - 102.18 + 60 \approx 9.7
   \]

**Answer for Part (b):** The minimum average cost per copy is approximately **9.7 cents**, and the corresponding number of copies made is approximately **7419**.

Let me know if you want any more details on the steps or have further questions. Here are some related questions to deepen your understanding:

1. How does the value of the leading coefficient affect the graph's shape and orientation?
2. What would change if the coefficient of \( x^2 \) were negative?
3. How is the vertex formula derived for a quadratic function?
4. What are some practical implications for businesses in minimizing average costs?
5. How would changes in production costs affect the average cost function?

**Tip:** Always verify the direction of a parabola by checking the sign of the \( x^2 \)-term's coefficient.

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 per copy using a quadratic function for a business producing color copies. This problem involves algebraic concepts like quadratic functions, minimization, and the vertex formula. Suitable for students in Grades 10-12.

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