Math CentralQuandaries & Queries


Question from Bruce, a teacher:

I figure that the surface area (SA) of a round bottom test tube can be determined by the equation SA = 2pi*h*r
where r is the radius of the hemispheric bottom & column part, and h is the height (h) at any point upwards from the bottommost portion in the center of the tube.

I haven't been able to come up with an equation yet for the circumference of such a test tube at any point upwards along the tube based on the height in its center or on the distance along the outside of the tube.

I would appreciate any help or thoughts - Thanks

Hi Bruce,

I got the same expression for the area in the answer to an earlier question about a contact lens. (I used R where you used r.) If you look at what I did you will see that h units up from the bottom of the test tube the radius of the circular cap is √(2Rh - h2) as long as h ≤ R. If h > R then the radius of the test tube is R. Since the circumference of a circle is 2 π r where r is the radius the circumference of the test tube at any point h units upward from the deepest point is

circumference = 2 π √(2Rh - h2) if h ≤ R and
circumference = 2 π R if h > R.



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