Thread: Ratio of Volumes of a cone

1. Ratio of Volumes of a cone

What is the ratio of the volumes of a cone and its model made to the scale 1:2, 1:3, 1:n?

i don't get it. should i get the value of n or what?

2. I interpret the problem as follows: Suppose you have two cones, where the side lenghts of the two cones are related by the ratio 1:2. What then is the ratio of the volumes of the two cones?

In other words, the side lenghts of the second cone are twice as long as the sidelenghts of the first cone. What then is the ratio of the volume of second cone compared to the volume of the first cone?

Then they ask the same question, if the side lenghts of the second cone are n times as long as the sidelenghts of the first cone, where n is some number.

3. Then the ratio of volumes would be the same right? Is it that simple?

4. The volume V of a cone is given by BH/3, where B is the area of the base of the cone, and where H is the height of the cone.

If cone no. 2 has twice the lenghts as those of cone no. 1, consider how this affects B and H, and combine these in the formula above.

5. Originally Posted by Dug
What is the ratio of the volumes of a cone and its model made to the scale 1:2, 1:3, 1:n?

i don't get it. should i get the value of n or what?
consider a basic cone with base radius r and height h

$\displaystyle V_1 = \dfrac{\pi}{3} r^2 h$

now double r and h ...

$\displaystyle V_2 = \dfrac{\pi}{3} (2r)^2 (2h) = 2^3 \cdot \dfrac{\pi}{3} r^2 h$

triple r and h and do the same as above ... what pattern do you see?

generalize the pattern for a cone with linear dimensions n times larger

6. oh i get it now.. so the answers will be:
1:2 = 1:8
1:3 = 1:27
1:n = 1:n³

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ration of cone & 1/27 part of cone

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