EugeneC's answer to yb ʕ •ᴥ•ʔ's Secondary 4 E Maths Singapore question.
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Hope that I’m correct ! Hahaha
Date Posted:
4 years ago
tysm !
although the method gives the correct values, the cutting shown in the diagram here is quite puzzling.
we have to consider that the bottle is actually 3D and the bottle has a long axis that is a vertical line through O. if you slice horizontally, you get circle cross-sections. if you slice diagonally, how do you get a hemisphere + (triangle/ pyramid/ cone?) with semicircle base?
I believe the correct way is to slice the top cap off horizontally along EF, then move this cap to the bottom to form a complete hemisphere. above XY, you are just left with a cylinder with radius OX and height XE.
by calculation, ...
volume of cylinder
= pi x r2 x h
= 71.086127
volume of hemisphere
= (1/2) x (4/3) x pi x r3
= 134.0412866
giving total volume = 205 cm3.
(amazingly, Eugene's method gives the same volumes for the 2 portions even though the shapes are hard to comprehend).
we have to consider that the bottle is actually 3D and the bottle has a long axis that is a vertical line through O. if you slice horizontally, you get circle cross-sections. if you slice diagonally, how do you get a hemisphere + (triangle/ pyramid/ cone?) with semicircle base?
I believe the correct way is to slice the top cap off horizontally along EF, then move this cap to the bottom to form a complete hemisphere. above XY, you are just left with a cylinder with radius OX and height XE.
by calculation, ...
volume of cylinder
= pi x r2 x h
= 71.086127
volume of hemisphere
= (1/2) x (4/3) x pi x r3
= 134.0412866
giving total volume = 205 cm3.
(amazingly, Eugene's method gives the same volumes for the 2 portions even though the shapes are hard to comprehend).
Actually, it is 1/2 of a cylinder instead of a triangle with a semicircle base. Woops
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