Calculate the sphericity of a cuboid having dimensions of 5A, 3A, and A.

Steps to Solve

1. 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.

2. Calculate the Volume of the Cuboid

The volume ( V ) of the cuboid with dimensions ( 5A, 3A, ) and ( A ) is calculated as:

$$ V = \text{Length} \times \text{Width} \times \text{Height} = 5A \times 3A \times A = 15A^3 $$

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,

$$ S = 2(15A^2 + 3A^2 + 5A^2) = 2(23A^2) = 46A^2 $$

4. Substitute the Volume and Surface Area into the Sphericity Formula

Now, substitute ( V ) and ( S ) into the sphericity formula:

$$ \phi = \frac{6 \sqrt{\pi} (15A^3)}{(46A^2)^2} $$

5. Simplify the Expression

Calculating ( (46A^2)^2 = 2116A^4 ):

$$ \phi = \frac{90 \sqrt{\pi} A^3}{2116 A^4} = \frac{90 \sqrt{\pi}}{2116A} $$