Calculate the sphericity of a cuboid having dimensions of 5A, 3A, and A.
Understand the Problem
The question is asking to calculate the sphericity of a cuboid with specific dimensions of 5A, 3A, and A. Sphericity is typically used to describe the roundness of an object compared to a sphere, which involves a specific formula.
Answer
The sphericity of the cuboid is $$ \phi = \frac{90 \sqrt{\pi}}{2116A} $$
Answer for screen readers
The sphericity of the cuboid is
$$ \phi = \frac{90 \sqrt{\pi}}{2116A} $$
Steps to Solve
- Identify the Sphericity Formula
The formula for sphericity $\phi$ is given by:
$$ \phi = \frac{6 \sqrt{\pi} V}{S^{2}} $$
where ( V ) is the volume of the object, and ( S ) is the surface area.
- Calculate the Volume of the Cuboid
The volume ( V ) of the cuboid with dimensions ( 5A, 3A, ) and ( A ) is calculated as:
$$ V = \text{Length} \times \text{Width} \times \text{Height} = 5A \times 3A \times A = 15A^3 $$
- Calculate the Surface Area of the Cuboid
The surface area ( S ) is calculated using the formula:
$$ S = 2(\text{Length} \times \text{Width} + \text{Width} \times \text{Height} + \text{Height} \times \text{Length}) $$
For the given dimensions:
$$ S = 2(5A \times 3A + 3A \times A + A \times 5A) $$
Calculating each term:
- ( 5A \times 3A = 15A^2 )
- ( 3A \times A = 3A^2 )
- ( A \times 5A = 5A^2 )
Thus,
$$ S = 2(15A^2 + 3A^2 + 5A^2) = 2(23A^2) = 46A^2 $$
- Substitute the Volume and Surface Area into the Sphericity Formula
Now, substitute ( V ) and ( S ) into the sphericity formula:
$$ \phi = \frac{6 \sqrt{\pi} (15A^3)}{(46A^2)^2} $$
- Simplify the Expression
Calculating ( (46A^2)^2 = 2116A^4 ):
$$ \phi = \frac{90 \sqrt{\pi} A^3}{2116 A^4} = \frac{90 \sqrt{\pi}}{2116A} $$
The sphericity of the cuboid is
$$ \phi = \frac{90 \sqrt{\pi}}{2116A} $$
More Information
Sphericity is a measure of how closely the shape of an object resembles that of a sphere. The sphericity of 1 indicates a perfect sphere, while lower values indicate less spherical shapes.
Tips
- Forgetting to square the surface area in the sphericity formula is a common mistake. Ensure correct application of the formula.
- Miscalculating dimensions or misinterpreting them as numbers instead of variables can lead to incorrect results. Double-check calculations to avoid these errors.
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