Cubes: Using a formula to calculate volume

Let's solve this problem step-by-step:

Step 1: Given the surface area $S = 486$, we know the formula for the surface area of a cube is:

• $S = 6a^2$

Step 2: We need to rearrange this formula to find $a$. The equation becomes:

• $a^2 = \frac{S}{6}$
• $a = \sqrt{\frac{S}{6}}$

Step 3: Substitute the given surface area into this equation:

$a = \sqrt{\frac{486}{6}}$

Step 4: Perform the division:

$a = \sqrt{81}$

Step 5: Calculate the square root:

$a = 9$

Now that we have found the side length, let's find the volume:

Step 6: Use the formula for the volume of a cube:

• $V = a^3$

Step 7: Substitute $a = 9$ into the volume formula:

$V = 9^3$

Step 8: Calculate the cube:

$V = 729$

Thus, the length of the side of the cube is $\mathbf{9}$ and the volume of the cube is $\mathbf{729}$.

The final answer matches the given multiple choice result:

$a=9,V=729$

To solve this problem, we will calculate the volume and surface area of a cube with edge length 8 cm.

• Step 1: Calculate the volume of the cube using the formula $V = a^3$.

Given that the edge length $a = 8$ cm, the volume is calculated as follows:

$V = 8^3 = 8 \times 8 \times 8 = 512 \text{ cm}^3$

• Step 2: Calculate the surface area of the cube using the formula $S = 6a^2$.

Using the same edge length $a = 8$ cm, we find the surface area:

$S = 6 \times 8^2 = 6 \times (8 \times 8) = 6 \times 64 = 384 \text{ cm}^2$

Thus, the calculated volume and surface area of the cube are, respectively, 512 cm$^3$ and 384 cm$^2$.

Therefore, the correct solution to the problem, matching the given answer choices, is choice 1: $V = 512, S = 384$.

To solve this problem, we'll calculate the volume of the cube using the formula for the volume of a cube, which is $V = a^3$, where $a$ is the length of an edge.

Step-by-step solution:

• Step 1: Identify the given length of the cube's edge. Here, $a = 5$ cm.
• Step 2: Apply the formula for the volume of a cube: $V = a^3$.
• Step 3: Substitute the value of $a$ into the formula: $V = 5^3$.
• Step 4: Calculate $5^3$. Since $5 \times 5 = 25$ and $25 \times 5 = 125$, we get $5^3 = 125$.

Therefore, the volume of the cube is $125$ cm$^3$.

Looking at the answer choices, the correct answer is choice 2: $125$ cm$^3$.

The volume of a cube is determined by the formula $V = s^3$, where $s$ is the length of each edge of the cube.

Given that each edge of the cube measures $4$ cm, we proceed as follows:

• Step 1: Identify the edge length as $s = 4$ cm.
• Step 2: Apply the formula $V = s^3$.
• Step 3: Calculate $V = 4^3 = 4 \times 4 \times 4$.

Continuing with the calculation:

• $4 \times 4 = 16$
• $16 \times 4 = 64$

The volume of the cube is therefore $64$ cubic centimeters.

Therefore, the solution to the problem is $\ 64$ cm³.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information.
• Step 2: Apply the appropriate formula for volume.
• Step 3: Perform the necessary calculations.

Now, let's work through each step:
Step 1: The problem states that each edge of the cube is 3 cm long.
Step 2: We use the formula for the volume of a cube: $V = s^3$, where $s$ is the side length.
Step 3: Substituting the side length into the formula gives us $V = 3^3$. Calculating this, we have:

$3^3 = 3 \times 3 \times 3 = 27$.

Therefore, the volume of the cube is $27$ cm³.

Given the answer choices, the correct choice is $\textbf{27 cm³}$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information - The edge length of the cube is 2 cm.
• Step 2: Apply the appropriate formula to find the volume of the cube.
• Step 3: Perform the calculation with the given edge length.

Now, let's work through each step:
Step 1: The problem provides that each edge of the cube measures 2 cm.
Step 2: The formula for the volume of a cube is $V = s^3$, where $s$ is the edge length.
Step 3: Plugging in the values, we calculate the volume as follows:

Therefore, the volume of the cube is $\mathbf{8 \, \text{cm}^3}$.

To solve this problem, we will follow a straightforward method.

Step 1: Identify the given information.
The problem states that the edge of the cube is 1 cm long.

Step 2: Apply the formula for the volume of a cube.
The volume $V$ of a cube with edge length $a$ is given by the formula: $V = a^3$

Step 3: Substitute the given edge length into the formula.
Substituting $a = 1$ cm into the formula gives: $V = 1^3$

Step 4: Perform the calculation.
Calculate $1^3$, which equals 1.

Therefore, the volume of the cube is $1$ cm³.

To solve this problem, we will follow these steps:

• Step 1: Identify the given information.

• Step 2: Apply the formula for the volume of a cube.

• Step 3: Perform the calculation to find the volume.

Let's work through these steps:

Step 1: The problem states a cube with edge length $a = 6$ cm.

Step 2: The formula to calculate the volume of a cube is $V = a^3$.

Step 3: Substitute the given edge length:

$V = 6^3$

$V = 6 \times 6 \times 6$

$V = 216$ cm³

Therefore, the volume of the cube is $216$ cm³.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information
• Step 2: Apply the appropriate formula
• Step 3: Perform the necessary calculations

Now, let's work through each step:

Step 1: Identify the given information.
The length of each edge of the cube is given as $7$ cm.

Step 2: Apply the appropriate formula.
The formula for the volume $V$ of a cube where each edge is of length $s$ is given by $V = s^3$.

Step 3: Perform the necessary calculations.
Plugging in the value of $s = 7$ cm, we compute the volume as follows:
$V = 7^3 = 7 \times 7 \times 7 = 343 \text{ cm}^3$

Therefore, the volume of the cube is $343$ cm³.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information
• Step 2: Apply the formula for the volume of a cube
• Step 3: Perform the necessary calculations to find the volume

Now, let's work through each step:
Step 1: The problem gives us the edge length of the cube as 8 cm.
Step 2: We'll use the formula for the volume of a cube, which is $V = s^3$, where $s$ is the edge length.
Step 3: Plugging the edge length into the formula, we have $V = 8^3$. This calculates to:
$V = 8 \times 8 \times 8 = 512$.

Therefore, the volume of the cube is $512$ cm³, which is option 2 from the choices given.

To solve this problem, we will calculate the volume of a cube using the standard formula for its volume:

• Volume formula of a cube: $V = s^3$, where $s$ is the side length.

Given that each edge of the cube is 0.5 cm, we substitute this value into the formula:

$V = (0.5 \, \text{cm})^3$

Calculating the cube of 0.5:

$0.5^3 = 0.5 \times 0.5 \times 0.5$

$= 0.25 \times 0.5$

$= 0.125 \, \text{cm}^3$

However, it is important to recognize that the calculations when simplified produce a fraction:

$0.125$ can be written in fraction form as $\frac{1}{8}$.

Therefore, the volume of the cube is $\frac{1}{8} \, \text{cm}^3$, which matches the given answer choice.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information.
• Step 2: Apply the appropriate formula for calculating the volume of a cube.
• Step 3: Perform the necessary calculation to find the volume.

Let's work through each step:
Step 1: We know the length of each edge of the cube is 10 cm.
Step 2: The formula for the volume $V$ of a cube with side length $s$ is given by $V = s^3$.
Step 3: Plugging the given side length into the formula, we get:
$V = 10^3 = 1000$ cm³.

Therefore, the volume of the cube is $1000$ cm³.

To find the volume of a cube, we use the formula:

$V = \text{side}^3$

Here, the side length of the cube is given as $2X$. Substituting this into the formula gives:

$V = (2X)^3$

To expand this expression, apply the cube to every part of the expression:

$V = 2^3 \times X^3$

$V = 8 \times X^3$

Thus, the volume of the cube is:

$V = 8X^3$

Therefore, the correct answer is:

$8X^3$

Thus, the solution to the problem is $8x^3$.