Question

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

Answer 1

Answer:

  (1/10)∛100 ≈ 0.4642

Step-by-step explanation:

For a cube of volume V, the edge length is ...

  s = ∛V

and the area is ...

  A = 6s² = 6V^(2/3)

Then the ratio of area to volume is ...

  r1 = A/V = 6V^(2/3)/V = 6V^(-1/3)

If the value of V is increased by a factor of 10, the ratio of area to volume is now ...

  r2 = 6(10V)^(-1/3)

The factor by which the ratio changed is ...

  r2/r1 = (6(10V)^(-1/3))/(6V^(-1/3)) = 10^(-1/3) = (1/10)∛100

The surface area per unit volume changes by a factor of (∛100)/10.

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Example

For a cube of side length 2, the volume is 2³ = 8 and the surface area is 6·2² = 24. The ratio of surface area to volume is 24/8 = 3.

Multiplying the edge length by ∛10, we now have a volume of (2∛10)³ = 80, and a surface area of 6(2∛10)² = 24(10^(2/3)). The ratio of surface area to volume is ... 24(10^(2/3))/80 = 0.3∛100.

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