Two cylinders are equal in volume. The radius of one is doubled, and the height of the other cylinder is increased to \( k \) times its original height. If the two new cylinders are equal in volume, what is the value of \( k \)?
Answer (2)
f i rs t cy l in d er : r a d i u s − 2 r 1 h e i g h t − h 1 seco n d cy l in d er : r a d i u s − r 1 h e i g h t − k ∗ h 1 V o l u m e 1 = V o l u m e 2 π ( 2 r 1 ) 2 ∗ h 1 = π r 1 2 ∗ k ∗ h 1 ∣ d i v i d e b y h 1 ∗ π 4 r 1 2 = k ∗ r 1 2 ∣ d i v i d e b y r 1 2 k = 4
To ensure the volumes of the new cylinders are equal, the value of k , which represents the factor by which the height of the second cylinder is increased, must be 4. This is derived from the established volume relationships of cylinders. When the radius of the first cylinder is doubled, the height of the second must increase by 4 times to maintain equal volume.
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