Maybe an impossible shape?

Geometry Level 3

There is a cone, a hemisphere and a cylinder standing on an equal base. Given that they have same heights and the heights are equal to the radius, their volumes would have a definite ratio.

If the ratio is a : b : c a : b : c , where a , b , c a,b,c are positive integers such that a + b + c a+b+c is minimized, find a b + b c − a c ab + bc - ac .


The answer is 5.

Skanda Prasad by Skanda Prasad , 20, India

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1 solution

Rishabh Jain
Feb 5, 2016

Volume of : ( A ) C o n e = V a = 1 3 π ( r 2 ) r = ( 1 3 ) π r 3 \color{#EC7300}{(A)Cone=V_a=\dfrac{1}{3}\pi(r^2)r}=\color{#20A900}{(\dfrac{1}{3})}\pi r^3 ( B ) H e m i s p h e r e = V c = 2 3 π ( r 3 ) = 2 3 π r 3 \color{darkviolet}{(B)Hemisphere=V_c=\dfrac{2}{3}\pi(r^3)}=\color{#20A900}{\dfrac{2}{3}}\pi r^3 ( C ) C y c l i n d e r = V b = π ( r 2 ) r = ( 1 ) π r 3 \color{#D61F06}{(C)Cyclinder=V_b=\pi(r^2)r}=\color{#20A900}{(1)}\pi r^3 Hence a:b:c= 1:2:3
ab+bc-ac=2+6-3= 5 \large\boxed5

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Cool! Same solution of mine...but I'm not so proficient at latex...

Skanda Prasad - 5 years, 4 months ago

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Neither I am... Keep practising and certainly you will be some day

Rishabh Jain - 5 years, 4 months ago

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Hmmm...yes, got to practice...

Skanda Prasad - 5 years, 4 months ago

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