Calculate Volume Of Right Rectangular Prism With Given Dimensions

Let's dive into understanding how to express the volume of a right rectangular prism, especially when we have some relationships defined between its dimensions. This is a classic problem in geometry, and it's super useful for all sorts of real-world applications. We will break down the problem step-by-step, making sure everyone can follow along. So, let's get started, guys!

Problem Statement

The problem gives us some key information about a right rectangular prism. Specifically, it tells us that:

• The height of the prism is three times the width of the base.
• The length of the base is twice the width.

Our mission, should we choose to accept it, is to find an expression that represents the volume of this prism using only one variable: w, which stands for the width of the base.

To solve this problem effectively, we will:

1. Recall the formula for the volume of a right rectangular prism.
2. Express the height and length in terms of the width, w.
3. Substitute these expressions into the volume formula.
4. Simplify the resulting expression to get our final answer.

Volume of a Right Rectangular Prism

The volume of any right rectangular prism (think of it as a box) is found by multiplying its length (l), width (w), and height (h). Mathematically, this is expressed as:

$$V = l \times w \times h$$

Where:

• V is the volume
• l is the length
• w is the width
• h is the height

This formula is fundamental, so make sure you've got it locked in your memory!

Expressing Dimensions in Terms of w

Now, let's translate the information given in the problem into mathematical expressions. This is where the problem starts to become less abstract and more concrete.

• Height: The problem states that the height is three times the width. So, we can write the height (h) as:

  $$h = 3w$$

• Length: Similarly, the length of the base is twice the width. We can express the length (l) as:

  $$l = 2w$$

So, we've now successfully expressed both the height and the length in terms of w. This is a crucial step because it allows us to substitute these expressions into the volume formula.

Substituting into the Volume Formula

Next up, we'll take these expressions for height and length and plug them into our volume formula. Remember, the volume formula is:

$$V = l \times w \times h$$

We know that l = 2w and h = 3w. So, let's substitute those in:

$$V = (2w) \times w \times (3w)$$

This substitution is a pivotal step. We've now got an expression for the volume that only involves the variable w. All that's left to do is simplify it.

Simplifying the Expression

Okay, guys, let’s simplify this expression. We have:

$$V = (2w) \times w \times (3w)$$

To simplify, we just multiply the terms together. First, let's multiply the constants: 2 and 3. That gives us 6.

Then, we multiply the w terms: w × w × w. This is the same as w raised to the power of 3, which we write as w³.

So, our simplified expression becomes:

$$V = 6w^3$$

And there we have it! The volume of the prism, expressed in terms of w, is 6w³. This is our final answer.

Common Mistakes to Avoid

When working through these types of problems, it’s easy to make a few common mistakes. Keep an eye out for these pitfalls to make sure you’re on the right track:

• Forgetting the Formula: The most common mistake is simply forgetting the formula for the volume of a rectangular prism. Always start by writing down the formula to keep it fresh in your mind.
• Incorrect Substitution: Make sure you’re substituting the expressions correctly. Double-check that you’ve matched the right expressions for length, width, and height.
• Simplification Errors: When simplifying, be careful with the exponents. Remember that w × w × w is w³, not 3w.
• Misinterpreting the Problem: Sometimes, the problem statement can be a bit tricky. Read it carefully and make sure you understand the relationships between the dimensions before you start solving.

Real-World Applications

Understanding how to calculate volumes is not just a theoretical exercise; it has tons of real-world applications. Think about:

• Construction: Architects and builders use volume calculations to determine how much material is needed for a project.
• Packaging: Companies need to know the volume of boxes and containers to ship their products efficiently.
• Fluid Dynamics: Engineers use volume calculations to design pipelines and storage tanks.
• Everyday Life: Even in everyday situations, like packing a suitcase or filling a fish tank, volume calculations come in handy.

So, the skills you’re learning here are not just for exams; they’re practical tools that you’ll use throughout your life.

Practice Problems

To really nail this concept, it’s important to practice. Here are a couple of practice problems you can try:

1. A right rectangular prism has a width that is one-fourth of its height, and its length is twice its width. Express the volume of the prism in terms of its height.
2. The length of a right rectangular prism is five times its width, and the height is half its length. If the width is x, what is the volume of the prism in terms of x?

Working through these problems will help you solidify your understanding and build confidence.

Conclusion

Alright, guys, we've covered a lot in this article. We've broken down how to find the volume of a right rectangular prism when given relationships between its dimensions. Remember the key steps:

1. Write down the volume formula: V = l × w × h.
2. Express the length and height in terms of the width, w.
3. Substitute these expressions into the formula.
4. Simplify the resulting expression.

By mastering these steps and avoiding common mistakes, you’ll be well-equipped to tackle any similar problem. Keep practicing, and you’ll become a pro at volume calculations in no time! This skill is super useful, and you'll find it popping up in many different areas of math and real life. So, keep up the great work!