The radii of the circular ends of a solid frustum of a cone are 33 cm and 27 cm, and its slant height is 10 cm. Find its capacity and total surface area.
Greater radius = R = 33 cm
Smaller radius = r = 27 cm
Slant height = l = 10 cm
Using the formula for height of a frustum:
Height = h =
$=\sqrt{l^{2}-(R-r)^{2}}$
$=\sqrt{10^{2}-(33-27)^{2}}$
$=\sqrt{100-(6)^{2}}$
$=\sqrt{100-36}$
$=\sqrt{64}=8 \mathrm{~cm}$
Capacity of the frustum
$=\frac{1}{3} \pi h\left(R^{2}+r^{2}+R r\right)$
$=\frac{1}{3} \times \frac{22}{7} \times 8\left(33^{2}+27^{2}+33 \times 27\right)$
$=\frac{22 \times 8}{3 \times 7} \times 2709=22704 \mathrm{~cm}^{3}$
Surface area of the frustum
$=\pi R^{2}+\pi r^{2}+\pi l(R+r)$
$=\pi\left[R^{2}+r^{2}+l(R+r)\right]$
$=\frac{22}{7}\left[33^{2}+27^{2}+10(33+27)\right]$
$=\frac{22}{7}[1089+729+10(60)]$
$=\frac{22 \times 2418}{7}=7599.43 \mathrm{~cm}^{2}$