SOLUTION: The two right circular cones are similar. The height of the larger cone is 80 cm, and the area of its base is 3675 cm squared. The area of the base of the smaller cone is 147 Cm sq

Algebra ->  Volume -> SOLUTION: The two right circular cones are similar. The height of the larger cone is 80 cm, and the area of its base is 3675 cm squared. The area of the base of the smaller cone is 147 Cm sq      Log On


   

Question 1151760: The two right circular cones are similar. The height of the larger cone is 80 cm, and the area of its base is 3675 cm squared. The area of the base of the smaller cone is 147 Cm squared. What is the volume of the smaller cone?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
One dimensional measurements (side lengths, perimeters, etc.) are in the ratio a%3Ab.
Two dimensional measurements (area, surface area, etc.) are in the ratio of a%5E2%3Ab%5E2
Three dimensional measurements (volume) are in the ratio of a%5E3%3Ab%5E3


The ratio of the smaller area to the larger area is 147%2F3675=1%2F25
then. the ratio of the height of the smaller cone and height of the larger cone
(80cm )is
h%2F80=sqrt%281%2F25%29=1%2F5 => h=16cm

so, ratio of the heights is h%5Bs%5D%2Fh%5Bl%5D=1%2F5

then, volumes are in the ratio of
a%5E3%2Fb%5E3=%281%2F5%29%5E3
a%5E3%2Fb%5E3=1%2F125
a%5E3%2Fb%5E3=0.008

using formula for the area of a smaller cone, we calculate radius:
A=++pi%2Ar%28r+%2B+sqrt%28r%5E2+%2B+16%5E2%29%29
147=++3.14%2Ar%28r+%2B+sqrt%28r%5E2+%2B16%5E2%29%29
r=2.5cm
using formula for the area of a larger cone, we calculate radius:
A+=++pi%2Ar%28r+%2B+sqrt%28r%5E2+%2B+h%5E2%29%29
3675=++pi%2Ar%28r+%2B+sqrt%28r%5E2+%2B+80%5E2%29%29
r=12.5cm

the volume of a smaller cone is:
V+=+%281%2F3%29pi%2Ar%5E2%2Ah=>V+=+%281%2F3%29pi%2A%282.5%29%5E2%2A16=>highlight%28V=104.72cm%5E3%29
the volume of a larger cone is:
V+=+%281%2F3%29pi%2Ar%5E2%2Ah=>V+=+%281%2F3%29pi%2A%2812.5%29%5E2%2A80=13089.97cm%5E3

check the ratio:
104.72%2F13089.97+=0.008