Volume Of A Right Rectangular Prism Calculation And Examples
This article delves into the concept of calculating the volume of a right rectangular prism, focusing on a specific scenario where the height is related to the base length. We will explore the fundamental principles of volume calculation and apply them to solve the given problem. This guide aims to provide a clear and comprehensive understanding of the topic, suitable for students and anyone interested in geometry and spatial reasoning. Our main keyword is volume of a right rectangular prism, so we will make sure this keyword is included in the beginning of the paragraphs.
Problem Statement
The problem presents a right rectangular prism with a square base. The edge length of the square base is given as x units, and the height of the prism is 3 units greater than the length of the base. The objective is to determine the expression that represents the volume of the prism in cubic units. The provided options are:
- A. $x^3+9$
- B. $x^3+3$
- C. $3x^2$
- D. $x^2+3x$
To solve this problem, we need to recall the formula for the volume of a right rectangular prism and apply it to the given dimensions.
Core Concepts Volume of a Right Rectangular Prism
The volume of a right rectangular prism is the amount of space it occupies. It is calculated by multiplying the area of the base by the height of the prism. In mathematical terms:
Volume = Base Area × Height
For a right rectangular prism with a rectangular base, the base area is simply the product of the length and width of the rectangle. If we denote the length as l, the width as w, and the height as h, then the volume V is given by:
V = l × w × h
In our specific case, the base is a square, which means the length and width are equal. If the edge length of the square base is x, then the base area is x × x = x². The height of the prism is given as 3 units greater than the length of the base, so the height is x + 3.
Applying the Formula
Now, we can apply the volume of a right rectangular prism formula to our problem. We have:
- Base Area = x²
- Height = x + 3
Therefore, the volume V of the prism is:
V = x² × (x + 3)
To find the expression that represents the volume, we need to expand this product:
V = x² * (x) + x² * (3)
V = x³ + 3x²
Step-by-Step Solution
Let's break down the solution process step by step to ensure clarity and understanding. Our main goal is to calculate the volume of a right rectangular prism.
Step 1 Identify the Given Information
We are given the following information:
- The base of the prism is a square.
- The edge length of the square base is x units.
- The height of the prism is 3 units greater than the length of the base, which means the height is x + 3 units.
Step 2 Determine the Base Area
Since the base is a square, its area is the square of its side length. The side length is x, so the base area is:
Base Area = x²
Step 3 Determine the Height
The height of the prism is given as 3 units greater than the length of the base. The length of the base is x, so the height is:
Height = x + 3
Step 4 Apply the Volume Formula
The volume of a right rectangular prism is the product of the base area and the height. So, we have:
Volume = Base Area × Height
Volume = x² × (x + 3)
Step 5 Expand the Expression
To find the expression that represents the volume, we need to expand the product:
Volume = x²(x) + x²(3)
Volume = x³ + 3x²
Step 6 Match the Expression with the Options
The expression we obtained for the volume is x³ + 3x². Comparing this with the given options, we find that it matches option D.
Therefore, the expression that represents the volume of the prism in cubic units is x³ + 3x².
Detailed Explanation of the Solution
The problem requires us to find an expression for the volume of a right rectangular prism given specific dimensions. The key to solving this problem lies in understanding the formula for the volume of a prism and applying it correctly.
The volume of a right rectangular prism is calculated by multiplying the area of its base by its height. In this case, the base is a square with side length x, so the area of the base is x². The height of the prism is given as x + 3. Therefore, the volume V can be expressed as:
V = Base Area × Height
V = x² × (x + 3)
To simplify this expression, we distribute x² across the terms inside the parentheses:
V = x² * (x) + x² * (3)
V = x³ + 3x²
This expression, x³ + 3x², represents the volume of the prism in cubic units. By comparing this result with the given options, we can identify the correct answer.
Common Mistakes to Avoid
When solving problems related to the volume of a right rectangular prism, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help in avoiding them.
Mistake 1 Incorrectly Calculating the Base Area
A common mistake is miscalculating the area of the base. In this case, the base is a square, so its area is the square of the side length. If the side length is x, the base area is x², not 2x or some other incorrect expression.
Mistake 2 Misinterpreting the Height
The problem states that the height is 3 units greater than the length of the base. This means the height is x + 3, not 3x or any other misinterpretation of the given information. Reading the problem statement carefully and understanding the relationships between the dimensions is crucial.
Mistake 3 Incorrectly Applying the Volume Formula
The volume of a right rectangular prism is the product of the base area and the height. A mistake can occur if these quantities are added instead of multiplied, or if the formula is otherwise misapplied. It’s essential to remember the correct formula: Volume = Base Area × Height.
Mistake 4 Errors in Expanding the Expression
After setting up the volume expression, expanding it correctly is crucial. In this case, the expression is x²(x + 3). The distributive property must be applied correctly to multiply x² by both x and 3. A common mistake is to multiply x² only by x or to incorrectly distribute the terms.
Mistake 5 Not Matching the Expression with the Options
After obtaining the expression for the volume, it is important to compare it carefully with the given options. Sometimes, the correct expression may be presented in a different form, and it's necessary to ensure they are equivalent. In this case, the final expression x³ + 3x² needs to be matched with the options provided.
Practice Problems
To reinforce your understanding of calculating the volume of a right rectangular prism, here are some practice problems:
- A right rectangular prism has a square base with an edge length of 5 units. The height of the prism is 2 units greater than the base length. Find the volume of the prism.
- The volume of a right rectangular prism with a square base is 108 cubic units. The height of the prism is 3 units. What is the length of the edge of the base?
- A right rectangular prism has a base with sides of length x and 2x. The height of the prism is x + 4. Write an expression for the volume of the prism.
Conclusion
Calculating the volume of a right rectangular prism is a fundamental concept in geometry. By understanding the formula and applying it systematically, you can solve a variety of problems. In the given problem, we determined that the expression representing the volume of the prism is x³ + 3x². Avoiding common mistakes and practicing with different problems will further enhance your understanding and problem-solving skills in this area.
