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If the surface area of a sphere and a cube are equal then how do you show that their volumes are in the ratio #sqrt(7) : sqrt(5)# ?

Answer 1

Savannah Anderson

This proposition is false...

Suppose the sphere has radius #r# and the cube side #t#.

Since the surface areas are equal, we have:

#4pi r^2 = 6t^2#

So #(r/t)^2 = 3/(2pi)# and #(r/t)^3 = (3/(2pi))^(3/2)#

Then:

#V_"sphere" / V_"cube"=(4/3 pi r^3)/t^3=4/3pi (r/t)^3 = 4/3pi (3/(2pi))^(3/2) ~~ 1.38197659788534191701#

Whereas:

#sqrt(7)/sqrt(5) ~~ 1.18321595661992320851#

So the proposition is false.

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Answer 2

Ethan Adkins

To show that the volumes of a sphere and a cube with equal surface areas are in the ratio √7 : √5, use the formulas for surface area and volume of each shape. Set the surface areas equal to each other and solve for the ratio of their volumes. The ratio of the volumes will be √7 : √5.

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