Question

A container, open at the top, is in the form of a frustum of a cone of height 24 cm with radii of its lower and upper circular ends as 8 cm and 20 cm, respectively. Find the cost of milk which can completely fill the container at the rate of ₹21 per litre. [CBSE 2014]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Height},h=24\mathrm{cm}, \\ \mathrm{Upper}\mathrm{radius},R=20\mathrm{cm}\mathrm{and} \\ \mathrm{Lower}\mathrm{radius},r=8\mathrm{cm} \\ \mathrm{Now}, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{container}=\frac{1}{3}\mathrm{\pi }h\left(R^2+r^2+Rr\right) \\ =\frac{1}{3}\times \frac{22}{7}\times 24\times \left({20}^2+8^2+20\times 8\right) \\ =\frac{176}{7}\times \left(400+64+160\right) \\ =\frac{176}{7}\times 624 \\ =\frac{109824}{7}{\mathrm{cm}}^3 \\ =\frac{109.824}{7}\mathrm{L}\left(\mathrm{As},1000{\mathrm{cm}}^3=1\mathrm{L}\right) \\ \mathrm{So},\mathrm{the}\mathrm{cost}\mathrm{of}\mathrm{the}\mathrm{milk}\mathrm{in}\mathrm{the}\mathrm{container}=\frac{109.824}{7}\times 21 \\ =329.472 \\ \approx ₹329.47 \end{gathered}$$

Hence, the cost of milk which can completely fill the container is ₹329.47.