Mensuration problem!

Geometry Level 4

The ratio of the volumes of the two cones is 4 : 5 4: 5 .

The ratio of the radii of their bases is 2 : 3 2: 3 .

Find the ratio of their slant heights.

6 : 4 6:4 Can't be determined 8 : 9 8:9 2 : 1 2:1

Adarsh Goyal by Adarsh Goyal , 29, India

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1 solution

L e t R 1 , H 1 a n d R 2 , H 2 b e t h e r a d i i a n d h e i g h t s o f t w o c o n e s . S o R 1 R 2 = 2 3 . R 1 2 ∗ H 1 R 2 2 ∗ H 2 = 4 ∗ H 1 9 ∗ H 2 = 4 5 . ⟹ H 1 H 2 = 9 5 . ∴ R 1 = 2 3 R 2 , H 1 = 9 5 H 2 . ∴ r a t i o o f s l a n t h e i g h t = R 1 2 + H 1 2 R 2 2 + H 2 2 . ∴ r a t i o o f s l a n t h e i g h t = 4 9 ∗ R 2 2 + 81 25 ∗ H 2 2 R 2 2 + H 2 2 . C a n n o t b e e v a l u a t e d f r o m g i v e n d a t a . Let~R_1,H_1 ~and~R_2,H_2~be ~the ~radii~and~heights~of~two~cones.\\ So~\dfrac{R_1}{R_2}=\dfrac 2 3.\\ \dfrac{R_1^2*H_1}{R_2^2*H_2}=\dfrac{4*H_1}{9*H_2}=\dfrac 4 5.\\ \implies~\dfrac{H_1}{H_2}=\dfrac 9 5.\\ \therefore~~R_1=\frac 2 3 R_2,~~~~~H_1=\frac 9 5 H_2.\\ \therefore~~ratio ~of~ slant~ height~=\dfrac { \sqrt{R_1^2+H_1^2} } { \sqrt{R_2^2+H_2^2} }.\\ \therefore~~ratio ~of~ slant~ height~=\dfrac { \sqrt{\frac 4 9 *R_2^2+\frac{81}{25} *H_2^2} } { \sqrt{R_2^2+H_2^2} }.\\ Can ~not~be~evaluated~from~given~data.

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