Consider a cube with an inscribed sphere. The function r times cap v is equal to cube root of 3 cap v over 4 pi can be used to find the radius, r , of a sphere given the volume, cap v The function s times r is equal to 2 r can be used to find the side length of the cube, s , depending on the radius of the sphere. The function cap A times s is equal to 6 s squared can be used to find the surface area, cap A , of the cube. Which is the composed function for the surface area of the cube for any given volume of an inscribed sphere?
Answer
A
cap A times open paren s times open paren r times cap v close paren close paren is equal to 12 times open paren cube root of 3 v over 4 pi close paren squared
B
cap A times open paren s times open paren r times cap v close paren close paren is equal to 24 times open paren cube root of 3 v over 4 pi close paren squared
C
cap A times open paren s times open paren r times cap v close paren close paren is equal to 24 times open paren th root of 3 v over 4 pi close paren cubed
D
cap A times open paren s times open paren r times cap v close paren close paren is equal to 12 times open paren th root of 3 v over 4 pi close paren cubed
Explanation
Substitute the radius function into the side-length function: s = 2r = 2∛(3v/(4π)). Then substitute that side length into the cube’s surface-area formula: A = 6s² = 6[2∛(3v/(4π))]² = 24(∛(3v/(4π)))². This matches choice B, the expression for the cube’s surface area in terms of the sphere’s volume.
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