A(x) = 7(x) + 600/x if a is 13 what is the average cost. how to reduce the cost to 15, what is the horizontal asymptote

Given the function \( A(x) = 7x + \frac{600}{x} \), which represents the average cost function:

### 1. **Find the average cost when \( x = 13 \):**
We substitute \( x = 13 \) into the function:

\[
A(13) = 7(13) + \frac{600}{13}
\]
\[
A(13) = 91 + \frac{600}{13}
\]
\[
A(13) = 91 + 46.15 \approx 137.15
\]

Thus, the average cost when \( x = 13 \) is approximately **137.15**.

### 2. **How to reduce the average cost to 15:**
To reduce the average cost to 15, we need to solve for \( x \) such that \( A(x) = 15 \):

\[
7x + \frac{600}{x} = 15
\]

Multiply through by \( x \) to eliminate the fraction:

\[
7x^2 + 600 = 15x
\]

Rearrange the equation into a standard quadratic form:

\[
7x^2 - 15x + 600 = 0
\]

Now we solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(7)(600)}}{2(7)}
\]
\[
x = \frac{15 \pm \sqrt{225 - 16800}}{14}
\]
\[
x = \frac{15 \pm \sqrt{-16575}}{14}
\]

Since the discriminant is negative, there is no real solution for this equation. Hence, it is **impossible** to reduce the average cost to 15 using real numbers.

### 3. **Horizontal asymptote:**
To find the horizontal asymptote of the function \( A(x) = 7x + \frac{600}{x} \), we examine the behavior of the function as \( x \to \infty \) or \( x \to -\infty \). As \( x \) becomes very large, the term \( \frac{600}{x} \) approaches zero.

Thus, the horizontal asymptote is given by the behavior of the linear term \( 7x \):

\[
\lim_{x \to \infty} A(x) = 7x
\]

Therefore, the function has **no horizontal asymptote** because the linear term \( 7x \) increases without bound.

### Recap:
- The average cost at \( x = 13 \) is approximately **137.15**.
- It is **impossible** to reduce the cost to 15 (no real solution).
- There is **no horizontal asymptote** since the linear term grows indefinitely.

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Would you like more details on any step?

Here are 5 related questions to explore further:
1. What is the interpretation of the vertical asymptote for this function?
2. How does the function behave as \( x \to 0 \)?
3. Can we find a minimum value for the average cost function?
4. What happens if we add a constant to the function? How would that affect asymptotes?
5. What if the cost function had been cubic instead of linear?

**Tip:** When solving quadratic equations, always check the discriminant first to see if real solutions exist.

This problem involves calculating the average cost function A(x) = 7x + 600/x when x = 13, exploring how to reduce the cost to 15, and determining the horizontal asymptote. Key mathematical concepts include quadratic equations, rational functions, and limits. Suitable for students in Grades 9-12.

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