Question

The radii of the circular ends of a bucket of height 15 cm are 14 cm and r cm (r < 14). If the volume of bucket is 5390 cm³, then find the value of r. [CBSE 2011]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Height},h=15\mathrm{cm}, \\ \mathrm{Radius}\mathrm{of}\mathrm{the}\mathrm{upper}\mathrm{end},R=14\mathrm{cm}, \\ \mathrm{Radius}\mathrm{of}\mathrm{lower}\mathrm{end}=r, \\ \mathrm{As}, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{bucket}=5390{\mathrm{cm}}^3 \\ \Rightarrow \frac{1}{3}\mathrm{\pi }h\left(R^2+r^2+Rr\right)=5390 \\ \Rightarrow \frac{1}{3}\times \frac{22}{7}\times 15\times \left({14}^2+r^2+14r\right)=5390 \\ \Rightarrow \frac{110}{7}\times \left(196+r^2+14r\right)=5390 \\ \Rightarrow 196+r^2+14r=\frac{5390\times 7}{110} \\ \Rightarrow 196+r^2+14r=343 \\ \Rightarrow r^2+14r-147=0 \\ \Rightarrow r^2+21r-7r-147=0 \\ \Rightarrow r\left(r+21\right)-7\left(r+21\right)=0 \\ \Rightarrow \left(r+21\right)\left(r-7\right)=0 \\ \Rightarrow r+21=0\mathrm{or}r-7=0 \\ \Rightarrow r=-21\mathrm{or}r=7 \\ \mathrm{As},r\mathrm{cannot}\mathrm{be}\mathrm{negative} \\ \therefore r=7\mathrm{cm} \end{gathered}$$

So, the value of r is 7 cm.