Question

A bucket made up of a metal sheet is in the form of a frustum of a cone of height 16 cm and radii of its lower and upper ends are 8 cm and 20 cm, respectively. Find the cost of the bucket if the cost of metal sheet used is Rs 15 per 100 cm².

Answer 1

Greater radius of the frustum = R = 20 cm
Smaller radius of the frustum = r = 8 cm
Height of the frustum = h = 16 cm

Slant height, l, of the frustum
$$\begin{gathered} =\sqrt{h^2+\left(R^2-r^2\right)} \\ =\sqrt{{16}^2+{\left(20-8\right)}^2} \\ =\sqrt{256+{\left(12\right)}^2} \\ =\sqrt{256+144} \\ =\sqrt{400}=20\mathrm{cm} \end{gathered}$$

Surface area of the frustum
$$\begin{gathered} ={\mathrm{\pi r}}^2+\mathrm{\pi l}\left(\mathrm{R}+\mathrm{r}\right) \\ =\mathrm{\pi }\left[r^2+l\left(R+r\right)\right] \\ =\frac{22}{7}\left[8^2+20\left(20+8\right)\right] \\ =\frac{22}{7}\left[64+20\left(28\right)\right] \\ =\frac{22\times 624}{7}=1961.14{\mathrm{cm}}^2 \end{gathered}$$

100 cm² of metal sheet costs Rs 15.
So, total cost of the metal sheet used for fabrication
$$\begin{gathered} =\frac{1961.14}{100}\times 15 \\ =\mathrm{Rs}294.171 \end{gathered}$$