Select The Correct Volume.Sarah Has A Solid Wooden Cube With A Length Of $\frac{4}{5}$ Centimeter. From Each Of Its 8 Corners, She Cuts Out A Smaller Cube With A Length Of $\frac{1}{5}$ Centimeter. What Is The Volume Of The Block

Mar 11, 2025 by ADMIN 234 views

**Select the Correct Volume: A Mathematical Conundrum**

Introduction

In mathematics, volume is a fundamental concept that measures the amount of space occupied by a three-dimensional object. When dealing with cubes, calculating volume is relatively straightforward, but things can get complicated when we introduce smaller cubes cut out from the corners. In this article, we will delve into the world of mathematics and explore how to find the volume of a block with smaller cubes removed from its corners.

The Problem

Sarah has a solid wooden cube with a length of $\frac{4}{5}$ centimeter. From each of its 8 corners, she cuts out a smaller cube with a length of $\frac{1}{5}$ centimeter. The problem asks us to find the volume of the block after the smaller cubes have been removed.

Understanding the Problem

To tackle this problem, we need to understand the concept of volume and how it applies to cubes. The volume of a cube is given by the formula:

V = s^3

where $s$ is the length of the side of the cube.

In this case, the original cube has a length of $\frac{4}{5}$ centimeter, so its volume is:

V_{original} = \left(\frac{4}{5}\right)^3 = \frac{64}{125}

Now, let's consider the smaller cubes that are cut out from the corners. Each smaller cube has a length of $\frac{1}{5}$ centimeter, so its volume is:

V_{smaller} = \left(\frac{1}{5}\right)^3 = \frac{1}{125}

Calculating the Volume of the Smaller Cubes

Since there are 8 smaller cubes, we need to find the total volume of all the smaller cubes. We can do this by multiplying the volume of a single smaller cube by 8:

V_{total\ smaller} = 8 \times \frac{1}{125} = \frac{8}{125}

Finding the Volume of the Block

Now that we have the volume of the original cube and the total volume of the smaller cubes, we can find the volume of the block by subtracting the volume of the smaller cubes from the volume of the original cube:

V_{block} = V_{original} - V_{total\ smaller}

Substituting the values we found earlier, we get:

V_{block} = \frac{64}{125} - \frac{8}{125} = \frac{56}{125}

Conclusion

In this article, we explored how to find the volume of a block with smaller cubes removed from its corners. We started by understanding the concept of volume and how it applies to cubes, and then we calculated the volume of the original cube and the total volume of the smaller cubes. Finally, we found the volume of the block by subtracting the volume of the smaller cubes from the volume of the original cube. The volume of the block is $\frac{56}{125}$ cubic centimeters.

Additional Tips and Tricks

When dealing with cubes, it's essential to remember that the volume of a cube is given by the formula $V = s^3$.
When calculating the volume of smaller cubes, make sure to multiply the volume of a single smaller cube by the number of smaller cubes.
When finding the volume of the block, subtract the volume of the smaller cubes from the volume of the original cube.

Frequently Asked Questions

Q: What is the volume of the original cube? A: The volume of the original cube is $\frac{64}{125}$ cubic centimeters.
Q: What is the volume of a single smaller cube? A: The volume of a single smaller cube is $\frac{1}{125}$ cubic centimeters.
Q: What is the volume of the block? A: The volume of the block is $\frac{56}{125}$ cubic centimeters.

References

[1] "Volume of a Cube" by Math Open Reference
[2] "Volume of a Cube" by Khan Academy

Glossary

Cube: A three-dimensional object with six square faces.
Volume: The amount of space occupied by a three-dimensional object.
Smaller cube: A cube with a length smaller than the original cube.
Block: A three-dimensional object with a volume smaller than the original cube.
Q&A: Select the Correct Volume =====================================

Introduction

In our previous article, we explored how to find the volume of a block with smaller cubes removed from its corners. We calculated the volume of the original cube, the total volume of the smaller cubes, and finally, the volume of the block. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the volume of the original cube?

A: The volume of the original cube is $\frac{64}{125}$ cubic centimeters.

Q: What is the volume of a single smaller cube?

A: The volume of a single smaller cube is $\frac{1}{125}$ cubic centimeters.

Q: What is the volume of the block?

A: The volume of the block is $\frac{56}{125}$ cubic centimeters.

Q: How do I calculate the volume of a cube?

A: To calculate the volume of a cube, you need to use the formula $V = s^3$, where $s$ is the length of the side of the cube.

Q: What is the difference between the volume of the original cube and the volume of the block?

A: The difference between the volume of the original cube and the volume of the block is the volume of the smaller cubes, which is $\frac{8}{125}$ cubic centimeters.

Q: Can I use a calculator to find the volume of the block?

A: Yes, you can use a calculator to find the volume of the block. Simply enter the values of the original cube and the smaller cubes, and the calculator will give you the volume of the block.

Q: What if I have a cube with a different length?

A: If you have a cube with a different length, you can use the same formula $V = s^3$ to find its volume. Simply substitute the length of the cube into the formula and calculate the result.

Q: Can I apply this method to find the volume of a block with smaller cubes removed from its corners in 3D space?

A: Yes, you can apply this method to find the volume of a block with smaller cubes removed from its corners in 3D space. However, you will need to use a more complex formula that takes into account the dimensions of the block and the smaller cubes.