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You are told that the shapes are "similar" in the mathematical sense, i.e. all distances change in proportion.greg_rack said:Homework Statement:: Statement attached below
Relevant Equations:: ##V=\frac{1}{3}\pi h(r_{1}^2+r_{1}r_{2}+r_{2}^2)##
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The question is: how do I know the increase in the minor and greater radius, given just the ratio of the heights?
Got it, thanks!haruspex said:You are told that the shapes are "similar" in the mathematical sense, i.e. all distances change in proportion.
Yes. If two objects are mathematically similar, all corresponding distances are in the same ratio, r:1 say. Then all corresponding areas are in the ratio r2:1 and all corresponding volumes are in the ratio r3:1.greg_rack said:is there a way to find the volume after the transformation without knowing the exact formula with all radiuses and stuff
The ratio of heights directly affects the change in volume. As the height ratio increases, the volume also increases proportionally. Similarly, as the height ratio decreases, the volume decreases proportionally.
No, the change in volume is not always proportional to the ratio of heights. Other factors such as the shape and size of the object can also impact the change in volume.
Yes, the change in volume can be negative even when the ratio of heights is positive. This can occur when the object undergoes a change in shape or when the ratio of heights is not the only factor affecting the volume.
The change in volume can be calculated by multiplying the original volume by the ratio of heights. For example, if the original volume is 10 cubic units and the ratio of heights is 2:1, the change in volume would be 20 cubic units.
Yes, understanding the change in volume given the ratio of heights is important in fields such as architecture, engineering, and construction. It can help in designing structures and calculating the volume of materials needed for construction projects.