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Question
If the areas of the adjacent faces of a rectangular block are in the ratio of 2:3:4 and it's volume is 9000 cm3 , then what is the length of the shortest side? And is the longest side as well?
If the areas of the adjacent faces of a rectangular block are in the ratio of 2:3:4 and it's volume is 9000 cm3 , then what is the length of the shortest side? And is the longest side as well?
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Solution
Let the edges (dimensions) of block be a, b, and c.
a*b : b*c : c*a = 2 : 3 : 4
a * b = 2 k b * c = 3 k c * a = 4 k for some k > 0
multiply all these :
a * b * b * c * c * a = a² b² c² = 24 k³
volume of block = a b c = 9, 000 cm³
Hence a² b² c² = 9, 000 * 9, 000 = 24 k³
k³ = 27,000,000/8
k = 300 / 2 = 150
Hence a * b = 300 b * c = 450 c * a = 600
a = abc / b c = 9000 / 450 = 20 cm
b = abc / ca = 9000 / 600 = 15 cm
c = abc / ab = 30 cm
so 15 cm is answer
a*b : b*c : c*a = 2 : 3 : 4
a * b = 2 k b * c = 3 k c * a = 4 k for some k > 0
multiply all these :
a * b * b * c * c * a = a² b² c² = 24 k³
volume of block = a b c = 9, 000 cm³
Hence a² b² c² = 9, 000 * 9, 000 = 24 k³
k³ = 27,000,000/8
k = 300 / 2 = 150
Hence a * b = 300 b * c = 450 c * a = 600
a = abc / b c = 9000 / 450 = 20 cm
b = abc / ca = 9000 / 600 = 15 cm
c = abc / ab = 30 cm
so 15 cm is answer
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