A solid cube is melted and half of its material is recast into a cuboid whose length, breadth, and height are in the ratio 2 : 1 : 1. The difference between the volume of the original cube and the cuboid formed is 250 cm3. Find the total surface area of the cuboid.
250 cm2
Let the volume of the original cube be \(V\). Half of the material is recast, so the cuboid volume is \(\frac{V}{2}\).
The difference in volumes is \(V - \frac{V}{2} = \frac{V}{2}\), and this is given as 250 cm3.
So \(\frac{V}{2} = 250\), giving \(V = 500\) and cuboid volume \(= 250\) cm3.
Let the dimensions be \(2x, x, x\) in the ratio \(2:1:1\). Then volume \(= 2x \cdot x \cdot x = 2x^3\).
So \(2x^3 = 250\), giving \(x^3 = 125\) and \(x = 5\).
The dimensions are length 10 cm, breadth 5 cm, height 5 cm.
Total surface area \(= 2(lb + bh + hl) = 2(10\cdot5 + 5\cdot5 + 10\cdot5) = 2(50+25+50) = 2 \times 125 = 250\) cm2.
Hence, the total surface area of the cuboid is 250 cm2.
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Important Questions from 3-D Mensuration
Find the total surface area of a closed cylinder having a base radius of 70 m and a height of 110 m. [Use π = \(22\over7\)]
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The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone.
(Use $\pi = \frac{22}{7}$)- A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$ - The volume (in $m^3$) of a cube, each of whose edges is 42 m, is:
