The volumes of two spheres are #729# #i n^3# and #27# #i n^3#. What is the ratio of their radii, rounded to the nearest whole number?

Volume of bigger sphere is# (4pi)/3*R^3 = 729# cubic in.

Volume of smaller sphere is# (4pi)/3*r^3 = 27# cubic in. Where #R and r# are thrir respective radius

#:.(cancel((4pi)/3)*R^3) /(cancel((4pi)/3)*r^3)=729/27 or (R/r)^3=27 or (R/r)^3=3^3 or R/r=3 :.R:r =3:1#

The ratio of their radii is #3:1# [Ans]

Vol. of a sphere=#4/3pir^3#
#4/3pir^3=729#
#4/3pir^3=27#
multiply both sides by#(1)/(4/3pi)#
#r^3=(27)/(4/3pi#
#r^3=27/4.188790205#
#r^3=6.445775195#
#r=root3(6.445775195)#
#r=1.861#
#r=±2#
#4/3pir^3=729#
multiply both sides by#(1)/(4/3pi)#
#r^3=(729)/(4/3pi#
#r^3=729/4.188790205#
#r^3=174.0359303#
#r=root3(174.0359303)#
#r=5.583# or
#r=±6#
# =2/6=2:6# ratio of their radii

Ratios are always in simplest form, so #1 : 3#

However we are given the ratios as
Big sphere : small sphere, so the ratio is #3 : 1#

All spheres are similar figures.

Their volumes are in the same ratio as the cubes of their radii.

#(R/r)^3 = 729/27 = 9^3/3^3#

Note that we are not asked to find the actual lengths of their radii, only the ratio.

Find the cube root of both sides.

#R/r = root3(729/27)#

#R/r = 9/3 = 3/1#

The ratio #R : r = 3:1#

Ratios are always given in the simplest form.