Tyler Has Two Cube-shaped Storage Spaces In His Apartment Building, One Large And One Small. The Small Storage Space Has A Volume Of $12 \, \text{ft}^3$. Tyler Wants To Know The Total Volume Of Both Storage Spaces.Let $s$ Be The

Introduction

In mathematics, understanding the properties of three-dimensional objects is crucial for solving various problems. Tyler, a resident of an apartment building, has two cube-shaped storage spaces - one large and one small. The small storage space has a volume of $12 \, \text{ft}^3$. In this article, we will delve into the concept of volume and dimensions of cube-shaped objects, and explore how to find the total volume of both storage spaces.

Understanding Volume and Dimensions

Volume is a measure of the amount of space inside a three-dimensional object. For a cube-shaped object, the volume can be calculated using the formula:

$$V = s^3$$

where $V$ is the volume and $s$ is the length of a side of the cube.

The Small Storage Space

The small storage space has a volume of $12 \, \text{ft}^3$. To find the length of a side of the cube, we can use the formula:

$$s = \sqrt[3]{V}$$

where $s$ is the length of a side of the cube and $V$ is the volume.

Plugging in the value of $V$, we get:

$$s = \sqrt[3]{12}$$

Using a calculator, we can find the value of $s$:

$$s \approx 2.28 \, \text{ft}$$

The Large Storage Space

Since the large storage space is also cube-shaped, we can assume that it has the same dimensions as the small storage space. However, we are not given the volume of the large storage space. To find the total volume of both storage spaces, we need to find the volume of the large storage space.

Finding the Volume of the Large Storage Space

Let's assume that the large storage space has a side length of $x$ feet. We can use the formula for the volume of a cube to find the volume of the large storage space:

$$V = x^3$$

However, we are not given the value of $x$. To find the value of $x$, we can use the fact that the large storage space has the same dimensions as the small storage space.

Using Similarity to Find the Volume of the Large Storage Space

Since the large storage space has the same dimensions as the small storage space, we can use the concept of similarity to find the volume of the large storage space. Similarity is a concept in geometry that states that if two figures have the same shape, but not necessarily the same size, then their corresponding sides are proportional.

In this case, we can set up a proportion to relate the side lengths of the two cubes:

$$\frac{s}{x} = \frac{2.28}{x} = \frac{2.28}{s}$$

where $s$ is the side length of the small storage space and $x$ is the side length of the large storage space.

Simplifying the proportion, we get:

$$x = \frac{2.28 \times s}{s}$$

Plugging in the value of $s$, we get:

$$x = \frac{2.28 \times 2.28}{2.28}$$

Simplifying the expression, we get:

$$x = 2.28 \times 2.28$$

Using a calculator, we can find the value of $x$:

$$x \approx 5.18 \, \text{ft}$$

Finding the Volume of the Large Storage Space

Now that we have the value of $x$, we can find the volume of the large storage space:

$$V = x^3$$

Plugging in the value of $x$, we get:

$$V = (5.18)^3$$

Using a calculator, we can find the value of $V$:

$$V \approx 140.51 \, \text{ft}^3$$

Finding the Total Volume of Both Storage Spaces

To find the total volume of both storage spaces, we can add the volumes of the small and large storage spaces:

$$V_{\text{total}} = V_{\text{small}} + V_{\text{large}}$$

Plugging in the values of $V_{\text{small}}$ and $V_{\text{large}}$, we get:

$$V_{\text{total}} = 12 + 140.51$$

Simplifying the expression, we get:

$$V_{\text{total}} = 152.51 \, \text{ft}^3$$

Conclusion

In this article, we explored the concept of volume and dimensions of cube-shaped objects. We used the formula for the volume of a cube to find the volume of the small storage space, and then used the concept of similarity to find the volume of the large storage space. Finally, we added the volumes of the small and large storage spaces to find the total volume of both storage spaces.

References

• [1] "Geometry" by Michael Artin
• [2] "Calculus" by Michael Spivak

Further Reading

• [1] "Mathematics for Computer Science" by Eric Lehman
• [2] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  Tyler's Cube-Shaped Storage Spaces: Q&A =============================================

Q: What is the volume of the small storage space?

A: The volume of the small storage space is $12 \, \text{ft}^3$.

Q: How do you find the length of a side of the cube?

A: To find the length of a side of the cube, you can use the formula:

$$s = \sqrt[3]{V}$$

where $s$ is the length of a side of the cube and $V$ is the volume.

Q: What is the length of a side of the small storage space?

A: Using the formula, we can find the length of a side of the small storage space:

$$s = \sqrt[3]{12}$$

Using a calculator, we can find the value of $s$:

$$s \approx 2.28 \, \text{ft}$$

Q: What is the volume of the large storage space?

A: We are not given the volume of the large storage space. However, we can assume that it has the same dimensions as the small storage space.

Q: How do you find the volume of the large storage space?

A: To find the volume of the large storage space, you can use the formula:

$$V = x^3$$

where $V$ is the volume and $x$ is the length of a side of the cube.

Q: How do you find the length of a side of the large storage space?

A: Since the large storage space has the same dimensions as the small storage space, we can use the concept of similarity to find the length of a side of the large storage space.

Q: What is the length of a side of the large storage space?

A: Using the concept of similarity, we can set up a proportion to relate the side lengths of the two cubes:

$$\frac{s}{x} = \frac{2.28}{x} = \frac{2.28}{s}$$

Simplifying the proportion, we get:

$$x = \frac{2.28 \times s}{s}$$

Plugging in the value of $s$, we get:

$$x = \frac{2.28 \times 2.28}{2.28}$$

Simplifying the expression, we get:

$$x = 2.28 \times 2.28$$

Using a calculator, we can find the value of $x$:

$$x \approx 5.18 \, \text{ft}$$

Q: What is the volume of the large storage space?

A: Now that we have the value of $x$, we can find the volume of the large storage space:

$$V = x^3$$

Plugging in the value of $x$, we get:

$$V = (5.18)^3$$

Using a calculator, we can find the value of $V$:

$$V \approx 140.51 \, \text{ft}^3$$

Q: What is the total volume of both storage spaces?

A: To find the total volume of both storage spaces, we can add the volumes of the small and large storage spaces:

$$V_{\text{total}} = V_{\text{small}} + V_{\text{large}}$$

Plugging in the values of $V_{\text{small}}$ and $V_{\text{large}}$, we get:

$$V_{\text{total}} = 12 + 140.51$$

Simplifying the expression, we get:

$$V_{\text{total}} = 152.51 \, \text{ft}^3$$

Q: What is the final answer?

A: The final answer is $152.51 \, \text{ft}^3$.