Select The Correct Answer.The Volume Of A Rectangular Prism Is A Minimum Of 25 Cubic Feet. The Height Of The Prism Is More Than The Width.Carla Wrote This System Of Inequalities To Represent This Situation, Where V V V Is The

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Introduction

In mathematics, a system of inequalities is a set of statements that contain one or more variables and are connected by the words "and," "or," or "not." These statements are often used to represent real-world problems, such as the dimensions of a rectangular prism. In this article, we will explore a system of inequalities that represents the volume of a rectangular prism, which is a minimum of 25 cubic feet, and the height of the prism is more than the width.

The Problem

The problem states that the volume of a rectangular prism is a minimum of 25 cubic feet. The height of the prism is more than the width. This can be represented mathematically as:

  • The volume of the prism is at least 25 cubic feet: V ≥ 25
  • The height of the prism is more than the width: h > w

Carla's System of Inequalities

Carla wrote the following system of inequalities to represent this situation:

  • V = lwh
  • V ≥ 25
  • h > w

Breaking Down the System of Inequalities

Let's break down each inequality in the system:

  • V = lwh: This equation represents the volume of the prism, which is equal to the product of its length, width, and height.
  • V ≥ 25: This inequality states that the volume of the prism is at least 25 cubic feet.
  • h > w: This inequality states that the height of the prism is more than the width.

Solving the System of Inequalities

To solve the system of inequalities, we need to find the values of l, w, and h that satisfy all three inequalities. We can start by substituting the first inequality into the second inequality:

lwh ≥ 25

Since h > w, we can substitute h = w + k, where k is a positive value:

l(w + k)w ≥ 25

Expanding the left-hand side of the inequality, we get:

lw^2 + kw^2 ≥ 25

Since l, w, and k are all positive values, we can divide both sides of the inequality by lw^2:

1 + (k/w) ≥ 25/lw

Now, we can substitute the third inequality into the inequality:

1 + (k/w) ≥ 25/lw h > w

Substituting h = w + k, we get:

w + k > w

Simplifying the inequality, we get:

k > 0

Since k is a positive value, we can conclude that:

k > 0

Conclusion

In this article, we explored a system of inequalities that represents the volume of a rectangular prism, which is a minimum of 25 cubic feet, and the height of the prism is more than the width. We broke down each inequality in the system and solved the system of inequalities to find the values of l, w, and h that satisfy all three inequalities. The solution to the system of inequalities is k > 0, which means that the height of the prism is more than the width.

Key Takeaways

  • A system of inequalities is a set of statements that contain one or more variables and are connected by the words "and," "or," or "not."
  • The volume of a rectangular prism is at least 25 cubic feet: V ≥ 25
  • The height of the prism is more than the width: h > w
  • Carla's system of inequalities is:
    • V = lwh
    • V ≥ 25
    • h > w
  • To solve the system of inequalities, we need to find the values of l, w, and h that satisfy all three inequalities.

Final Answer

Introduction

In our previous article, we explored a system of inequalities that represents the volume of a rectangular prism, which is a minimum of 25 cubic feet, and the height of the prism is more than the width. We broke down each inequality in the system and solved the system of inequalities to find the values of l, w, and h that satisfy all three inequalities. In this article, we will answer some frequently asked questions about the system of inequalities.

Q: What is the purpose of the system of inequalities?

A: The system of inequalities is used to represent the volume of a rectangular prism, which is a minimum of 25 cubic feet, and the height of the prism is more than the width.

Q: How do I write a system of inequalities?

A: To write a system of inequalities, you need to identify the variables and the relationships between them. In this case, the variables are l, w, and h, and the relationships are:

  • The volume of the prism is at least 25 cubic feet: V ≥ 25
  • The height of the prism is more than the width: h > w

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to find the values of the variables that satisfy all the inequalities. In this case, we substituted the first inequality into the second inequality and solved for k:

1 + (k/w) ≥ 25/lw

We then substituted h = w + k and simplified the inequality to get:

k > 0

Q: What is the solution to the system of inequalities?

A: The solution to the system of inequalities is k > 0, which means that the height of the prism is more than the width.

Q: Can I use the system of inequalities to find the dimensions of a rectangular prism?

A: Yes, you can use the system of inequalities to find the dimensions of a rectangular prism. However, you need to make sure that the values of l, w, and h satisfy all three inequalities.

Q: How do I check if the values of l, w, and h satisfy all three inequalities?

A: To check if the values of l, w, and h satisfy all three inequalities, you need to substitute them into each inequality and check if they are true. For example, if you have l = 5, w = 3, and h = 4, you can substitute these values into each inequality:

  • V = lwh: 5(3)(4) = 60
  • V ≥ 25: 60 ≥ 25 (true)
  • h > w: 4 > 3 (true)

Since all three inequalities are true, the values of l, w, and h satisfy the system of inequalities.

Q: Can I use the system of inequalities to find the maximum volume of a rectangular prism?

A: Yes, you can use the system of inequalities to find the maximum volume of a rectangular prism. To do this, you need to maximize the value of V = lwh while satisfying the other two inequalities.

Conclusion

In this article, we answered some frequently asked questions about the system of inequalities for a rectangular prism. We explained how to write a system of inequalities, how to solve a system of inequalities, and how to check if the values of l, w, and h satisfy all three inequalities. We also discussed how to use the system of inequalities to find the dimensions of a rectangular prism and the maximum volume of a rectangular prism.

Key Takeaways

  • A system of inequalities is a set of statements that contain one or more variables and are connected by the words "and," "or," or "not."
  • The volume of a rectangular prism is at least 25 cubic feet: V ≥ 25
  • The height of the prism is more than the width: h > w
  • Carla's system of inequalities is:
    • V = lwh
    • V ≥ 25
    • h > w
  • To solve the system of inequalities, you need to find the values of l, w, and h that satisfy all three inequalities.

Final Answer

The final answer is k > 0.