Two Cones
How to solve a truncated cone formula with two unknowns?
I have a foam coffee cup (you know the one you get from coffee shops which is a truncated cone) with top radius of 47.5mm, bottom radius of 28mm and height of 100mm, it has a volume of 457.65mL, i need to make the cup 1 and a half times bigger but keeping it in proportion, so just making it taller but adding half the volume onto the top, how would i go about this, please help, its for my maths assignment, thanks
It is a general principle that in any formula for the volume V of a solid body that depends upon parameters representing length, multiplying each of the length parameters by a fixed number α will increase the volume V by a factor of α^3.
For example, the volume of a box is given by V=lwh, where l, w, and h are the length, width, and height. If each of these parameters is multiplied by the same factor α (keeping the same proportions), then V will increase by a factor of α^3: (αl)(αw)(αh) = α^3V.
One more example: The volume of a sphere is given by V = (4/3)πr^3. If the length parameter r is multiplied by the factor α, then V will increase by a factor of α^3: (4/3)π(αr)^3 = α^3V.
This same principle will hold for your truncated cone. I do not even need to know the formula V for the volume of the truncated cone in order to know that if I increase each length parameter in the formula by a factor of α, then the volume will increase by a factor of α^3.
So your question really boils down to “By what factor α do I multiply each length parameter in the formula for the volume of the truncated cone, in order to increase the volume by a factor of 3/2?” Notice that multiplying each length parameter by some α is exactly how you keep the cone in proportion.
The volume increases by the factor α^3, so you want α to satisfy α^3=3/2. So α=(3/2)^(1/3)=1.14471.
Two-Cones Figure 8 Practice
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