Question 6
Derive the formula for the curved surface area and total surface area of the frustum of cone.

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Solution

Let ABC be a cone. A frustum DECB is cut by a plane parallel to its base. Let $r_{1} a n d r_{2}$ be the radii of the end of the frustum of the cone and h be the height of the frustum of the cone.
In $Δ A B G a n d Δ A D F, D F | | B G$
$∴ Δ A B G - Δ A D F$
$\frac{D F}{B G} = \frac{A F}{A G} = \frac{A D}{A B}$
$\frac{f^{2}}{r^{1}} = \frac{h_{1} - h}{h_{1}} = \frac{l_{1} - l}{l_{1}}$
$\frac{r_{2}}{r_{1}} = 1 - \frac{h}{h_{1}} = 1 - \frac{l}{l_{1}}$
$1 - \frac{h}{h_{1}} = \frac{r_{2}}{r_{1}}$
$\frac{h}{h_{1}} = 1 - \frac{r_{2}}{r_{1}} = \frac{r_{1} - r_{2}}{r_{1}}$
$\frac{h_{1}}{h} = \frac{r_{1}}{r_{1} - r_{2}}$
$h_{1} = \frac{r_{1} h}{r_{1} - r_{2}}$
Volume of frustum of cone = Volume of cone ABC - Volume of cone ADE

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