Mr. Ewald decides to make and sell Left Handed Smoke Shifters as a side business. The function below gives the average cost (in dollars) per Smoke Shifter when x Smoke Shifters are produced.

�(�)=4�+450�

Use the function to answer the following questions.

Determine A(19), and complete the sentence explaining the meaning of your answer. Round your answer to the nearest cent.

The average cost to produce 19 Left Handed Smoke Shifters is $.

How many Smoke Shifters must be produced in order to reduce the average cost to $13 each?

To reduce the average cost to produce each Smoke Shifter down to 13,  Smoke Shifters must be produced.

Give the equation of the horizontal asymptote of A(x)

Horizontal Asymptote:   

Select a sentence to explain its significance in this situation.

The function provided is $ A(x) = \frac{4x + 450}{x} $, which represents the average cost to produce $ x $ Left Handed Smoke Shifters.

### 1. **Determine $ A(19) $**:
We need to calculate the average cost for producing 19 Smoke Shifters.

\[
A(19) = \frac{4(19) + 450}{19} = \frac{76 + 450}{19} = \frac{526}{19} \approx 27.68
\]

**Meaning**: The average cost to produce 19 Left Handed Smoke Shifters is approximately **$27.68**.

### 2. **How many Smoke Shifters must be produced to reduce the average cost to $13 each?**
We need to solve for $ x $ when $ A(x) = 13 $.

\[
A(x) = \frac{4x + 450}{x} = 13
\]

Multiply both sides by $ x $:

\[
4x + 450 = 13x
\]

Simplify:

\[
450 = 9x
\]

Solve for $ x $:

\[
x = \frac{450}{9} = 50
\]

**Answer**: To reduce the average cost to $13, **50 Smoke Shifters** must be produced.

### 3. **Equation of the horizontal asymptote**:
The horizontal asymptote of the function $ A(x) = \frac{4x + 450}{x} $ can be found by analyzing the behavior of $ A(x) $ as $ x \to \infty $.

As $ x $ becomes very large, the term $ \frac{450}{x} $ approaches 0, and the function approximates:

\[
A(x) \approx 4
\]

Thus, the horizontal asymptote is:

\[
\text{Horizontal Asymptote: } y = 4
\]

### 4. **Significance of the horizontal asymptote**:
The horizontal asymptote $ y = 4 $ represents the minimum possible average cost per Smoke Shifter as production increases. In this situation, it means that no matter how many Smoke Shifters are produced, the average cost will approach $4, but it will never drop below this value.

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Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. What would the average cost be for producing 100 Smoke Shifters?
2. How does the average cost behave as production increases indefinitely?
3. What is the fixed cost component in this function, and how does it affect production costs?
4. How would a change in the initial production cost affect the horizontal asymptote?
5. What is the marginal cost of producing additional Smoke Shifters?

**Tip**: The horizontal asymptote shows the long-term behavior of a cost function. Always consider this when planning large-scale production.

Mr. Ewald decides to make and sell Left Handed Smoke Shifters as a side business. The function below gives the average cost (in dollars) per Smoke Shifter when x Smoke Shifters are produced. A(x)=4x+450/x. Use the function to answer the following questions. Determine A(19), and complete the sentence explaining the meaning of your answer. Round your answer to the nearest cent. The average cost to produce 19 Left Handed Smoke Shifters is $. How many Smoke Shifters must be produced in order to reduce the average cost to $13 each? To reduce the average cost to produce each Smoke Shifter down to 13, Smoke Shifters must be produced. Give the equation of the horizontal asymptote of A(x). Horizontal Asymptote:

Explore how to calculate the average cost of producing Left Handed Smoke Shifters using the function A(x) = (4x + 450) / x. Learn to determine the cost for 19 units, find the production needed to reduce the cost to $13, and understand the significance of the horizontal asymptote at y = 4. Suitable for Grades 9-12.

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