A thin metallic sheet, 1.92 m² in area, is cut into two equal pieces. One piece is used to make a hollow cube of volume P (in m³) and the other is used to make a hollow cuboid of volume Q (in m³) with dimensions in the ratio \(4:2:1\). If no sheet is wasted, which one of the following is correct?
\(343Q^2 = 216P^2\)
Each piece of sheet has area \(0.96\text{ m}^2\). For the hollow cube of edge a, the surface area gives \(6a^2 = 0.96\), so \(a^2 = 0.16\), and its volume is \(P = a^3\).
For the hollow cuboid with sides \(4k, 2k, k\), the surface area gives \(2(8k^2+2k^2+4k^2) = 28k^2 = 0.96\), so \(k^2 = 0.96/28\), and its volume is \(Q = 8k^3\).
Dividing the two surface-area relations, \(\dfrac{a^2}{k^2} = \dfrac{0.96/6}{0.96/28} = \dfrac{28}{6} = \dfrac{14}{3}\), so \(a^6 = \left(\dfrac{14}{3}\right)^3 k^6\), i.e. \(P^2 = \dfrac{2744}{27}k^6\). Since \(Q^2 = 64k^6\), this gives \(P^2 = \dfrac{2744}{27\times 64}Q^2 = \dfrac{343}{216}Q^2\), i.e. \(343Q^2 = 216P^2\).
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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 = \(22\over7\)]
- The curved surface area of a cylinder is half of its total surface area. If its height is 195 cm, then find its diameter (in cm).
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:
