A cube & Cuboid have the same volume the dimensions of the cuboid are in the ratio $1 : 2 : 4$ . If the difference between the cost of painting the cuboid and cube (whole surface area) at the rate of $R s 5 p e r s q . m t s$ is $R s 80$ . find their volumes.

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Solution

Let each side of cube be $^{'} a^{'}$ m.
Ratio of sides of cuboid is $1 : 2 : 4$ .
Let the sides be $^{'} x^{'}$ m
Volume of cube $=$ volume of cuboid
$∴ a^{3} = x \times 2 x \times 4 x = 8 x^{3} a^{3} = {8 x}^{3} \Rightarrow a = 2 x A / Q [6 a^{2} - 2 (l b + l h + b h)] \times 5 = 80 {6 \times {(2 x)}^{2} + 2 [{2 x}^{2} + {8 x}^{2} + {4 x}^{2}]} \times 5 = 80 \Rightarrow [- 24 x^{2} + 2 (14 x^{2})] \times 5 = 80 \Rightarrow (- 24 x^{2} + 28 x^{2}) \times 5 = 80 \Rightarrow 4 x^{2} \times 5 = 80 \Rightarrow 20 x^{2} = 80 \Rightarrow x^{2} = 4 \Rightarrow x = 2 ∴ a = 2 \times 2 = 4$
$∴$ Volume of cube $= {(4)}^{3} = 64 m^{3}$
sides of cuboid $= x, 2 x, 4 x = 2, 4, 8 m$
Volume of cuboid $= 2 \times 4 \times 8 = 64 m^{3}$

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