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Question 931541: Two spheres M and N have a volume of 36 PI cubic cm and 972 PI respectively. What is the ratio of their radii?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the volume of the smaller sphere is 36 * pi.
the volume of the larger sphere is 972 * pi.
the formula for the volume of a sphere is equal to 4/3 * pi * r^3
solve this formula for r and you get r = cube root of (volume / (4/3 * pi)) which can be simplified to (3*v) / (4*pi).
the radius of the smaller sphere is equal to 2.04835219
the radius of the larger sphere is equal to 6.145056569
the ratio of the radius of the smaller sphere to the radius of the larger sphere is therefore equal to 1/3.
there's an easier way to do this.
the ratio of any corresponding side of two similar figures is equal to the cube root of the ratio of their volumes.
if the figures are similar, then the ratio of any of their corresponding sides is the same.
with a circle, the corresponding sides would be the radius and the circumference.
the ratio of the volume of the smaller sphere to the volume of the larger sphere is equal to 36 / 972 which is equal to 1/27.
the cube root of 1/27 is equal to 1/3.
the ratio of any of the corresponding sides must be equal to 1/3.
i have already determined that the radii are in the ratio of 1/3.
what remains is to examine the circumference.
what also remains is to examine the the surface area.
the circumference of the smaller sphere is equal to 2*pi*r1
the circumference of the larger sphere is equal to 2 * pi * r2
the ratio of the circumference of the smaller sphere to the larger sphere is therefore equal to (2*pi*r1) / (2*pi*r2).
the 2*pi in the numerator and the denominator cancel out and you are left with r1/r2 which we already know is the ratio of 1/3.
the ratio of the surface area of the sphere should be equal to the square of the ratio of the sides.
that would make the ratio of the surface area equal to 1/9.
ratio of the sides is 1/3.
radio of the surface areas is 1/9
ratio of the volumes is 1/27.
the formula for the surface area of the sphere is 4*pi*r^2
once again, the ratio of the surface area of sphere1 and sphere2 becomes (4*pi*r1^2) / (4*pi*r2^2) which simplifies to r1^2/r2^2 which becomes 1/9.
the general rules are:
if the two figures are similar, then the ratio of their corresponding sides is the same.
the ratio of their corresponding areas is equal to the square of the ratio of their corresponding sides.
the ratio of their corresponding volumes is equal to the cube of the ratio of their corresponding sides.
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