Question

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.

Answer 1

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 =
$$\begin{gathered} =\sqrt{l^2-{\left(R-r\right)}^2} \\ =\sqrt{{10}^2-{\left(33-27\right)}^2} \\ =\sqrt{100-{\left(6\right)}^2} \\ =\sqrt{100-36} \\ =\sqrt{64}=8\mathrm{cm} \end{gathered}$$

Capacity of the frustum
$$\begin{gathered} =\frac{1}{3}\mathrm{\pi }h\left(R^2+r^2+Rr\right) \\ =\frac{1}{3}\times \frac{22}{7}\times 8\left({33}^2+{27}^2+33\times 27\right) \\ =\frac{22\times 8}{3\times 7}\times 2709=22704{\mathrm{cm}}^3 \end{gathered}$$

Surface area of the frustum

$$\begin{gathered} =\mathrm{\pi }{R}^2+\mathrm{\pi }{r}^2+\mathrm{\pi }l\left(R+r\right) \\ =\mathrm{\pi }\left[R^2+r^2+l\left(R+r\right)\right] \\ =\frac{22}{7}\left[{33}^2+{27}^2+10\left(33+27\right)\right] \\ =\frac{22}{7}\left[1089+729+10\left(60\right)\right] \\ =\frac{22\times 2418}{7}=7599.43{\mathrm{cm}}^2 \end{gathered}$$