How many square metres of canvas (approximately) will be required to make a conical tent 3 m high so that a man 2 m tall may stand anywhere within a radius of 1 m from its centre without stooping?
\(40\ \text{m}^2\)
Let the base radius of the cone be \(R\) m. Since the apex height is 3 m, the canvas height at a horizontal distance \(r\) from the axis is \(h(r)=3\left(1-\dfrac{r}{R}\right)\). For a 2 m tall man to stand upright at the edge of the 1 m radius region, \(h(1)=2\), i.e. \(3\left(1-\dfrac{1}{R}\right)=2\Rightarrow R=3\) m. The slant height is \(l=\sqrt{R^2+3^2}=\sqrt{18}=3\sqrt{2}\) m. Canvas required = curved surface area \(=\pi R l=9\sqrt{2}\,\pi\approx 40\ \text{m}^2\).
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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:
