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in Surface Areas And Volumes by (49.2k points)
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There are two prisms, one has equilateral triangle as a base and the other has a regular hexagon as a base. If both the prisms have equal height and volume, the ratio of length of each side of the equilateral triangle to the length of each side of the regular hexagon is 

(a) √3 :1 

(b) √6 :1 

(c) 3 : 2 

(d) 2 : √3

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Best answer

(b) \(\sqrt6 : 1\)

Let the length of each side of the equilateral triangle be a cm. 

Also, let the length of each side of the regular hexagon be b cm. 

Area of equilateral triangle = \(\frac{\sqrt3}{4}a^2\)

Area of regular hexagon = \(6\times\frac{\sqrt3}{4}b^2=\frac{3\sqrt3}{2}b^2\)

Since, volume of prism with equilateral triangle base = Volume of prism with hexagonal base 

\(\frac{\sqrt3}{4}a^2\times{h}=\frac{3\sqrt3}{2}b^2\times{h}\)

⇒ \(\frac{a^2}{b^2}=\frac{3\sqrt2}{2}÷\frac{\sqrt3}{4}=\frac61\) ⇒ a : b = \(\sqrt6 : 1\).

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