To solve the problem, we need to find the volume of a cube whose surface area is twice that of a cube with a volume of 1 unit³. Let’s break this down step by step:

Step 1: Surface Area and Volume Formulas

For a cube:

• The volume $V$ is given by $V = s^3$, where $s$ is the side length.
• The surface area $A$ is given by $A = 6s^2$.

Step 2: Consider the Cube with Volume of 1 Unit³

The volume of the first cube is given as $V_1 = 1$ unit³. Using the volume formula: $s_1^3 = 1 \implies s_1 = 1$ So, the side length of the first cube is $s_1 = 1$.

Now, calculate the surface area of this cube: $A_1 = 6s_1^2 = 6(1)^2 = 6 \text{ square units}.$

Step 3: Cube with Surface Area Twice the First Cube

The surface area of the second cube is twice that of the first cube: $A_2 = 2A_1 = 2 \times 6 = 12 \text{ square units}.$ Using the surface area formula $A_2 = 6s_2^2$ for the second cube: $12 = 6s_2^2 \implies s_2^2 = \frac{12}{6} = 2 \implies s_2 = \sqrt{2}.$

Step 4: Find the Volume of the Second Cube

Now, using the side length $s_2 = \sqrt{2}$ to calculate the volume of the second cube: $V_2 = s_2^3 = (\sqrt{2})^3 = 2\sqrt{2}.$

Thus, the volume of the second cube is $2\sqrt{2}$ cubic units.

Do you want further clarification or details on any part of this process?

Here are some related questions for you:

1. How do you calculate the surface area of other geometric shapes?
2. What is the relationship between volume and surface area for spheres?
3. Can the same method be applied for rectangular prisms?
4. How does changing the side length of a cube affect its surface area and volume?
5. What is the ratio of surface area to volume for the two cubes in this problem?

Tip: When solving problems with both surface area and volume, always connect the two formulas through the side length to simplify the solution.