SOLUTION: The area of a cross-section parallel to the base of a cone is 30, and the area of the base of the cone is 67.5. If the altitude of the cone is 6, then find the ratio of volumes be

Algebra ->  Bodies-in-space -> SOLUTION: The area of a cross-section parallel to the base of a cone is 30, and the area of the base of the cone is 67.5. If the altitude of the cone is 6, then find the ratio of volumes be      Log On


   

Question 1158312: The area of a cross-section parallel to the base of a cone is 30, and the area of the base of the cone is 67.5. If the altitude of the cone is 6, then find the ratio of volumes between the top cone and the frustum.
Found 2 solutions by ankor@dixie-net.com, greenestamps:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Area of a cross-section parallel to the base of a cone is 30, and the area of the base of the cone is 67.5.
If the altitude of the cone is 6, then find the ratio of volumes between the top cone and the frustum.
:
Find the total volume of the cone
V = 1%2F3*67.5*6
V = 135 cu/units
:
Find the radius of the base
R^2 = 67.5%2Fpi
R^2 = 21.486
R = sqrt%2821.486%29
R = 4.6353 is the Radius
:
Find the radius of the cross section
r^2 = 30%2Fpi
r^2 = 9.549
r = 3.09 is the small radius
:
The volume of a frustrum
V = 1.0472*h(R^2 + Rr + r^2)
V = 1.0472*h(21.486 + (4.6353*3.09) + 9.549)
V = 1.0472*h*45.358
V = 47.5h is the vol of the frustrum
:
The height of the small cone = (6-h)
Volume of the small cone
v = 1%2F3pi%2A9.549%286-h%29
v = 10(6-h)
v = 60 - 10h is the volume of the small cone
:
Vol of frustrum + vol of the cone = 135
47.5h + (60-10h) = 135
47.5h - 10h = 135 - 60
37.5h = 75
h = 75/37.5
h = 2 is the height of the frustrum
then
6-2 = 4 is the height of the small cone above the frustrum
:
The volume of the cone
v = 1%2F3pi%2A3.09%5E2%2A4
v = 40 cu/units
the volume of the frustrum
V = 47.5h
V = 47.5*2
V = 95 cu/units
:
"find the ratio of volumes between the top cone and the frustum."
40%2F95 = 8%3A19

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


I didn't follow all of the calculations in the solution shown by the other tutor.... But his answer is correct.

HOWEVER, he did WAY too much work getting the answer.

Here is a much easier way, using a basic principle of similar figures.

The top cone and the original cone are similar figures.

The ratio of the area of the cross section (base of the top cone) to the area of the base of the cone is 30:67.5 = 60:135 = 4:9 = 2^2:3^2.

That means the scale factor between the top cone and the original cone is 2:3, and the ratio between the volumes of the two cones is 8:27.

That in turn means the ratio of the volume of the top cone to the volume of the frustum is 8:19.

ANSWER: 8:19