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10 hours ago

If the volume of a cube is increased by a factor of 10, by what factor does the surface area per unit volume change

Mathematics

1 answer:

tester [3.9K]10 hours ago

8 0

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.

___

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.

The area-to-volume ratio changed by a factor of (0.3∛100)/3 = (∛100)/10, as previously noted.

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Check the attached figure for further clarification

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