The Cost, C C C , To Produce B B B Baseball Bats Per Day Is Modeled By The Function C ( B ) = 0.06 B 2 − 7.2 B + 390 C(b) = 0.06b^2 - 7.2b + 390 C ( B ) = 0.06 B 2 − 7.2 B + 390 . What Number Of Bats Should Be Produced To Keep Costs At A Minimum?A. 27 Bats B. 60 Bats C. 174 Bats D. 390 Bats

Introduction

In the world of business and economics, understanding the cost of production is crucial for making informed decisions. The cost function, denoted by $C(b)$, represents the total cost of producing $b$ units of a product. In this case, we are interested in finding the number of baseball bats that should be produced to keep costs at a minimum. To achieve this, we will use the given cost function $C(b) = 0.06b^2 - 7.2b + 390$ and apply mathematical techniques to determine the optimal production level.

Understanding the Cost Function

The cost function $C(b) = 0.06b^2 - 7.2b + 390$ is a quadratic function, which means it has a parabolic shape. The graph of this function is a U-shaped curve that opens upwards. This indicates that the cost of producing baseball bats increases as the number of bats produced increases.

Finding the Minimum Cost

To find the minimum cost, we need to find the vertex of the parabola represented by the cost function. The vertex of a parabola is the point where the function reaches its minimum or maximum value. In this case, we are interested in the minimum value.

The x-coordinate of the vertex of a parabola represented by the function $f(x) = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. In our case, $a = 0.06$ and $b = -7.2$. Plugging these values into the formula, we get:

$x = -\frac{-7.2}{2(0.06)}$ $x = -\frac{-7.2}{0.12}$ $x = 60$

Therefore, the x-coordinate of the vertex is 60. This means that the minimum cost occurs when 60 bats are produced.

Verifying the Result

To verify our result, we can plug $b = 60$ into the cost function to find the minimum cost:

$C(60) = 0.06(60)^2 - 7.2(60) + 390$ $C(60) = 0.06(3600) - 432 + 390$ $C(60) = 216 - 432 + 390$ $C(60) = 174$

This confirms that the minimum cost occurs when 60 bats are produced, and the minimum cost is $174.

Conclusion

In conclusion, the number of bats that should be produced to keep costs at a minimum is 60. This is determined by finding the vertex of the parabola represented by the cost function. By plugging $b = 60$ into the cost function, we verified that the minimum cost is indeed $174.

Answer

The correct answer is B. 60 bats.

Discussion

This problem illustrates the importance of understanding the cost function in business and economics. By analyzing the cost function, we can determine the optimal production level to minimize costs. This is a crucial decision-making tool for businesses and organizations.

Additional Resources

For more information on quadratic functions and cost functions, please refer to the following resources:

• Quadratic Functions
• Cost Functions

Related Problems

• Finding the Maximum Value of a Quadratic Function
• Optimizing Production with a Cost Function
  The Cost of Producing Baseball Bats: A Mathematical Approach ===========================================================

Q&A: The Cost of Producing Baseball Bats

Q: What is the cost function, and how is it used in business and economics? A: The cost function, denoted by $C(b)$, represents the total cost of producing $b$ units of a product. It is a crucial tool in business and economics, as it helps decision-makers determine the optimal production level to minimize costs.

Q: What type of function is the cost function, and what does its graph look like? A: The cost function is a quadratic function, which means it has a parabolic shape. The graph of this function is a U-shaped curve that opens upwards, indicating that the cost of producing baseball bats increases as the number of bats produced increases.

Q: How do you find the minimum cost using the cost function? A: To find the minimum cost, you need to find the vertex of the parabola represented by the cost function. The x-coordinate of the vertex is given by the formula $x = -\frac{b}{2a}$. In our case, $a = 0.06$ and $b = -7.2$. Plugging these values into the formula, we get $x = 60$.

Q: What does the x-coordinate of the vertex represent in the context of the problem? A: The x-coordinate of the vertex represents the number of bats that should be produced to keep costs at a minimum. In this case, the x-coordinate of the vertex is 60, which means that 60 bats should be produced to minimize costs.

Q: How do you verify the result by plugging $b = 60$ into the cost function? A: To verify the result, we plug $b = 60$ into the cost function to find the minimum cost:

$C(60) = 0.06(60)^2 - 7.2(60) + 390$ $C(60) = 0.06(3600) - 432 + 390$ $C(60) = 216 - 432 + 390$ $C(60) = 174$

This confirms that the minimum cost occurs when 60 bats are produced, and the minimum cost is $174.

Q: What is the minimum cost, and how is it related to the number of bats produced? A: The minimum cost is $174, and it occurs when 60 bats are produced. This means that producing 60 bats results in the lowest possible cost.

Q: What are some real-world applications of the cost function in business and economics? A: The cost function has numerous real-world applications in business and economics, including:

• Determining the optimal production level to minimize costs
• Evaluating the impact of changes in production levels on costs
• Comparing the costs of different production methods
• Making informed decisions about resource allocation

Q: What are some common mistakes to avoid when working with the cost function? A: Some common mistakes to avoid when working with the cost function include:

• Failing to consider the shape of the cost function
• Ignoring the impact of changes in production levels on costs
• Failing to verify results using the cost function
• Not considering the real-world implications of the cost function

Q: What are some additional resources for learning more about the cost function and its applications? A: Some additional resources for learning more about the cost function and its applications include:

• Quadratic Functions
• Cost Functions
• Optimizing Production with a Cost Function

Conclusion

In conclusion, the cost function is a powerful tool in business and economics, helping decision-makers determine the optimal production level to minimize costs. By understanding the cost function and its applications, individuals can make informed decisions about resource allocation and optimize production levels to achieve their goals.