A cube and a rectangular solid are equal in volume. If the lengths of the edges o fthe rectangular solid are 4, 8, and 16, what is the length of an edge of the cube?

Answer

To find the length of an edge of the cube given that it has the same volume as a rectangular solid with dimensions 4, 8, and 16, we can follow these steps:

### Step 1: Calculate the volume of the rectangular solid.
The volume \( V \) of a rectangular solid (or cuboid) is given by the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
For the given dimensions:
\[ V = 4 \times 8 \times 16 \]

### Step 2: Perform the multiplication.
First, multiply 4 and 8:
\[ 4 \times 8 = 32 \]
Now, multiply the result by 16:
\[ 32 \times 16 = 512 \]
So, the volume of the rectangular solid is:
\[ V = 512 \text{ cubic units} \]

### Step 3: Set the volume of the cube equal to the volume of the rectangular solid.
The volume \( V \) of a cube is given by:
\[ V = E^3 \]
where \( E \) is the length of an edge of the cube. Since the volumes are equal:
\[ E^3 = 512 \]

### Step 4: Solve for \( E \).
To find \( E \), take the cube root of both sides:
\[ E = \sqrt[3]{512} \]

### Step 5: Calculate the cube root.
We know that:
\[ 512 = 8^3 \]
Thus:
\[ E = 8 \]

### Conclusion:
The length of an edge of the cube is \( 8 \) units.

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