Answer:
(1/10)∛100 ≈ 0.4642
Step-by-step explanation:
For a cubic shape with volume V, the edge length is defined as...
s = ∛V
and the surface area is given by...
A = 6s² = 6V^(2/3)
The area-to-volume ratio is therefore...
r1 = A/V = 6V^(2/3)/V = 6V^(-1/3)
When the volume V is increased by a factor of 10, the new area-to-volume ratio becomes...
r2 = 6(10V)^(-1/3)
Consequently, the change factor for the ratio is...
r2/r1 = (6(10V)^(-1/3))/(6V^(-1/3)) = 10^(-1/3) = (1/10)∛100
The change in surface area per unit volume results in a factor of (∛100)/10.
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Example
For a cube with side length 2, the corresponding volume equals 2³ = 8, with a surface area of 6·2² = 24. The resulting area-to-volume ratio is 24/8 = 3.
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The area-to-volume ratio changed by a factor of (0.3∛100)/3 = (∛100)/10, as previously noted.
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