A room is L m long, B m wide and H m high. Further, L > B > H and L, B and H are integers. The length of the longest pole that can be placed in the room is 17 m and the length of the longest pole that can be placed on the floor is 15 m. What is the volume of the room?
\(864\ \text{m}^3\)
Let the room have dimensions \(L,B,H\) with \(L>B>H\). The space diagonal (longest pole in the room) gives \(L^2+B^2+H^2=17^2=289\), and the floor diagonal (longest pole on the floor) gives \(L^2+B^2=15^2=225\). Subtracting, \(H^2=64\Rightarrow H=8\). Since \(L^2+B^2=225\) with integers \(L>B>8\), the only solution is \(L=12,B=9\) (as \(12^2+9^2=144+81=225\)). Volume \(=L\times B\times H=12\times9\times8=864\ \text{m}^3\).
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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:
