Understanding the Domain of the Profit Formula

Oct 13, 2025 by wbrum.org 14 views

Introduction

In the world of business and manufacturing, profit is a crucial factor that determines the success of a company. The profit formula, which is a mathematical equation that calculates the profit based on various factors, is an essential tool for companies to make informed decisions. In this article, we will explore the profit formula used by a company to calculate the profit based on the number of tons of steel manufactured at the plant.

The Profit Formula

The profit formula used by the company is given by the equation:

p(t) = -3t^2 + 15t + 25

where $p(t)$ represents the profit and $t$ represents the number of tons of steel manufactured at the plant.

Understanding the Domain of the Profit Formula

To determine the appropriate values to use for $t$ , we need to understand the domain of the profit formula. The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of $t$ for which the profit formula is valid.

Analyzing the Profit Formula

To analyze the profit formula, we need to examine the coefficients of the quadratic equation. The coefficient of the squared term, $-3$ , is negative, which means that the parabola opens downward. This indicates that the profit function has a maximum value.

Finding the Maximum Value

To find the maximum value of the profit function, we need to find the vertex of the parabola. The vertex of a parabola is given by the equation:

t = -\frac{b}{2a}

where $a$ and $b$ are the coefficients of the quadratic equation. In this case, $a = -3$ and $b = 15$ . Plugging these values into the equation, we get:

t = -\frac{15}{2(-3)} = \frac{15}{6} = 2.5

This means that the maximum value of the profit function occurs when $t = 2.5$ .

Determining the Domain

Since the profit function has a maximum value, it is not defined for all real numbers. In fact, the domain of the profit function is restricted to the interval $[0, 2.5]$ . This means that the company can only manufacture steel up to 2.5 tons to maximize its profit.

Conclusion

In conclusion, the appropriate values to use for $t$ are all non-negative real numbers, specifically the interval $[0, 2.5]$ . This is because the profit function has a maximum value, and the company can only manufacture steel up to 2.5 tons to maximize its profit.

Recommendations

Based on the analysis of the profit formula, the company should consider the following recommendations:

Manufacture steel up to 2.5 tons: To maximize profit, the company should manufacture steel up to 2.5 tons.
Monitor and adjust production: The company should continuously monitor its production levels and adjust them as needed to ensure that it stays within the optimal range.
Consider alternative production methods: If the company finds that it is unable to manufacture steel up to 2.5 tons due to technical or logistical constraints, it should consider alternative production methods that can help it maximize its profit.

Final Thoughts

Introduction

In our previous article, we explored the profit formula used by a company to calculate the profit based on the number of tons of steel manufactured at the plant. We analyzed the domain of the profit formula and determined that the appropriate values to use for $t$ are all non-negative real numbers, specifically the interval $[0, 2.5]$ . In this article, we will answer some frequently asked questions about the profit formula and provide additional insights to help companies make informed decisions about their production levels.

Q&A

Q: What is the profit formula used by the company?

A: The profit formula used by the company is given by the equation:

p(t) = -3t^2 + 15t + 25

where $p(t)$ represents the profit and $t$ represents the number of tons of steel manufactured at the plant.

Q: What is the domain of the profit formula?

A: The domain of the profit formula is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of $t$ for which the profit formula is valid. Based on our analysis, the domain of the profit formula is the interval $[0, 2.5]$ .

Q: Why is the domain of the profit formula restricted to the interval $[0, 2.5]$ ?

A: The domain of the profit formula is restricted to the interval $[0, 2.5]$ because the profit function has a maximum value. This means that the company can only manufacture steel up to 2.5 tons to maximize its profit.

Q: What are the implications of the domain of the profit formula?

A: The implications of the domain of the profit formula are that the company should manufacture steel up to 2.5 tons to maximize its profit. Additionally, the company should continuously monitor its production levels and adjust them as needed to ensure that it stays within the optimal range.

Q: Can the company manufacture steel at levels above 2.5 tons?

A: No, the company should not manufacture steel at levels above 2.5 tons. This is because the profit function has a maximum value, and manufacturing steel at levels above 2.5 tons would result in a decrease in profit.

Q: What are some alternative production methods that the company could consider?

A: If the company finds that it is unable to manufacture steel up to 2.5 tons due to technical or logistical constraints, it should consider alternative production methods that can help it maximize its profit. Some possible alternative production methods include:

Increasing production capacity: The company could consider increasing its production capacity to manufacture more steel at lower costs.
Improving production efficiency: The company could consider improving its production efficiency to reduce waste and increase productivity.
Diversifying its product line: The company could consider diversifying its product line to include other steel products that can be manufactured at lower costs.

Q: How can the company ensure that it stays within the optimal range?

A: The company can ensure that it stays within the optimal range by continuously monitoring its production levels and adjusting them as needed. This can be done by:

Tracking production data: The company should track its production data to identify trends and patterns.
Analyzing production costs: The company should analyze its production costs to identify areas for improvement.
Making data-driven decisions: The company should make data-driven decisions to ensure that it stays within the optimal range.

Conclusion

In conclusion, the profit formula used by the company is a powerful tool that can help it make informed decisions about its production levels. By understanding the domain of the profit formula and analyzing the coefficients of the quadratic equation, the company can determine the appropriate values to use for $t$ and maximize its profit. We hope that this Q&A guide has provided additional insights and answered some frequently asked questions about the profit formula.