Question

A tent is made in the form of a conic frustum surmounted by a cone. The diameters of the base and the top of the frustum are $20m$ and $6m$ respectively and the height is $24m$. If the height of the tent is $28m$, find the quantity of canvas required.

Answer 1

Let $h$ be the height of the frustum and $r_1$ and $r_2$ be the radii of its circular bases.

We have $h=24m,r_1=10m,r_2=3m$

$l$= slant height of the frustum

$\Rightarrow l=\sqrt{{(r_1-r_2)}^2+h^2}=\sqrt{{(10-3)}^2+{24}^2}=25m$

for cone VA'B', we have
$l_2=$ slant height $=\sqrt{{O^{\prime }{B}^{\prime }}^2+{V{O}^{\prime }}^2}=\sqrt{3^2+4^2}=5m$

Quantity of canvas required
$=$ lateral surface area of frustum + lateral surface area of cone $V{A}^{\prime }{B}^{\prime }$

$=\pi (r_1-r_2)l+\pi r_2{l}_2$

$=\{\pi (10+3)\times 25+\pi \times 3\times 5\}{m}^2$

$=(325\pi +15\pi )m^2$

$=340\pi m^2$