Cube Surface Area 486: Calculate Side Length and 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:

$a=9,V=729$