A cube has a volume of 64 cubic metres. What is the length of its diagonal?
Understand the Problem
The question is asking to find the length of the diagonal of a cube given its volume. To solve this, we first need to determine the side length of the cube from the volume, and then use the formula for the diagonal of a cube.
Answer
$$ d = V^{1/3} \sqrt{3} $$
Answer for screen readers
The formula for the diagonal of the cube in terms of its volume $V$ is:
$$ d = V^{1/3} \sqrt{3} $$
Steps to Solve
- Finding the side length from volume
The volume $V$ of a cube can be expressed as ( V = s^3 ), where ( s ) is the side length of the cube. To find the side length, we rearrange the formula:
$$ s = V^{1/3} $$
- Calculating the diagonal of the cube
The diagonal ( d ) of a cube can be found with the formula ( d = s\sqrt{3} ). After calculating the side length, we can substitute it into this formula:
$$ d = s\sqrt{3} $$
- Combining the equations
Substituting ( s ) from the first step into the diagonal formula gives:
$$ d = (V^{1/3})\sqrt{3} $$
- Final expression for the diagonal
Now we have a final formula to calculate the diagonal of the cube in terms of its volume:
$$ d = V^{1/3} \sqrt{3} $$
The formula for the diagonal of the cube in terms of its volume $V$ is:
$$ d = V^{1/3} \sqrt{3} $$
More Information
Knowing the volume of a cube allows you to derive the side length easily, and the diagonal can be a useful measure in various applications, including geometry and architecture. The relationship between volume, side length, and diagonal showcases the three-dimensional properties of cubes.
Tips
- Misapplying the volume formula: Ensure to use ( V = s^3 ) correctly by isolating ( s ) properly.
- Forgetting to use ( \sqrt{3} ) when calculating the diagonal. It’s easy to skip the diagonal step thinking only about the side length.
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