miketopgun
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- Oct 6, 2010
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why is the surface area of a cube not the derivative of the volume of the cube
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surface area of cube not derivitive of volume
- Thread starter miketopgun
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lookagain said:miketopgun said:why is the surface area of a cube not the derivative of the volume of the cube
Rewrite: Why is the expression for the surface area of a cube not equal to the derivative
of the expression for the volume of the cube?
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An edge of a cube is more analogous to the diameter of a sphere instead of the radius, r.
Write the volume of a cube in terms of the diameter, d.
\(\displaystyle r = \frac{d}{2}\)
\(\displaystyle V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (\frac{d}{2})^3 = \frac{\pi d^3}{6}\)
\(\displaystyle V' = \frac{\pi d^2}{2}\)
The surface area of the sphere is \(\displaystyle 4 \pi r^2 = 4 \pi (\frac{d}{2})^2 = \pi d^2\)
For example, here the derivative of the expression for volume in terms of the diameter is
not equal to the expression for the surface area in terms of the diameter.
So you were comparing an edge of a cube with the different type of dimension of the
radius of a sphere when you took derivatives of their respective volume expressions.
And that was to try to get the expressions for their respective surface areas.