Question

If the radii of the circular ends of a bucket 28 cm high, are 28 cm and 7 cm, then find its capacity and total surface area. [CBSE 2011]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Height},h=28\mathrm{cm}, \\ \mathrm{Radius}\mathrm{of}\mathrm{the}\mathrm{upper}\mathrm{end},R=28\mathrm{cm}\mathrm{and} \\ \mathrm{Radius}\mathrm{of}\mathrm{the}\mathrm{lower}\mathrm{end},r=7\mathrm{cm} \\ \mathrm{Also}, \\ \mathrm{The}\mathrm{slant}\mathrm{height},l=\sqrt{{\left(R-r\right)}^2+h^2} \\ =\sqrt{{\left(28-7\right)}^2+{28}^2} \\ =\sqrt{{21}^2+{28}^2} \\ =\sqrt{441+784} \\ =\sqrt{1225} \\ =35\mathrm{cm} \\ \mathrm{Now}, \\ \mathrm{Capacity}\mathrm{of}\mathrm{the}\mathrm{bucket}=\frac{1}{3}\mathrm{\pi }h\left(R^2+r^2+Rr\right) \\ =\frac{1}{3}\times \frac{22}{7}\times 28\times \left({28}^2+7^2+28\times 7\right) \\ =\frac{88}{3}\times \left(784+49+196\right) \\ =\frac{88}{3}\times 1029 \\ =30184{\mathrm{cm}}^3 \\ \mathrm{Also}, \\ \mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{bucket}=\mathrm{\pi }l\left(R+r\right)+\mathrm{\pi }{r}^2 \\ =\frac{22}{7}\times 35\times \left(28+7\right)+\frac{22}{7}\times 7\times 7 \\ =110\times \left(35\right)+154 \\ =3850+154 \\ =4004{\mathrm{cm}}^2 \end{gathered}$$

Disclaimer: The answer of the total surface area given in the textbook is incorrect. The same has been corrected above.