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Question
Two cubes have volumes in the ratio 1 : 64. The ratio of the area of a face of first cube to that of the other is ______.
Options
1 : 4
1 : 8
1 : 16
1 : 32
Solution
Two cubes have volumes in the ratio 1 : 64. The ratio of the area of a face of first cube to that of the other is 1 : 16.
Explanation:
Let a and b be the edges of the two cubes, respectively.
Then, according to the question,
a3 : b3 = 1 : 64 ...[∵ Volume of cube = (edge)3]
⇒ `a^3/b^3 = 1/64`
⇒ `(a/b)^3 = (1/4)^3`
⇒ `a/b = 1/4` ...[Taking cube roots on both sides]
Now, ratio of areas, `(a/b)^2 = (1/4)^2` ...[∵ Surface area of cube = 6 × (edge)2]
⇒ `a^2/b^2 = 1/16`
∴ a2 : b2 = 1 : 16
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