A cylinder has a radius of 3 cm and a length of 10 cm. Every dimension of the cylinder is multiplied by 3 to form a new cylinder. How is the ratio of the volumes related to the ratio of the corresponding dimensions?
Answer (2)
V 1 = π r a d i u s 1 2 l e n g t h 1 r a d i u s 1 = 3 c m l e n g t h 1 = 10 c m V 1 = π 3 2 ∗ 10 = 90 π c m 3 V 2 = π ( 3 r a d i u s 1 ) 2 ∗ 3 l e n g t h 1 V 2 = π 9 2 ∗ 30 = 2430 π c m 3 R a t i o = V 2 V 1 = 2430 90 = 27 1 R a t i o : 1 : 27.
When the dimensions of the cylinder are multiplied by 3, the volume increases by a factor of 27. The original volume is 90π cm³, and the new volume is 2430π cm³. Therefore, the ratio of the volumes is 1:27, showing that volume scales with the cube of the linear dimensions.
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