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A solid wooden toy is in the form of hemisphere surmounted by a cone of same radius. The radius of hemisphere is 3.5 cm and the total wood used in the making of toy is \[166\frac{5}{6}c{{m}^{3}}\]. Find the height of the toy. Also, find the cost of painting hemispherical part of the toy at the rate of \[Rs.\,\,10\,\,per\,\,c{{m}^{2}}\]. \[\left[ \text{Use}\,\,\pi =\frac{22}{7} \right]\]

Answer:

Given, the radius of hemisphere is 3.5 cm and let the height of the cone be h cm.

Now,    Volume of wood \[=166\frac{5}{6}c{{m}^{3}}\]

\[\frac{2}{3}\pi {{r}^{3}}+\frac{1}{3}\pi {{r}^{2}}h=166\frac{5}{6}\]

\[\frac{2}{3}\times \frac{22}{7}\times 3.5\times 3.5\times 3.5+\frac{1}{3}\times \frac{22}{7}\times 3.5\times 3.5\times h\]

\[=\frac{1001}{6}\]

\[\frac{22}{7}\times 3.5\times 3.5\left( \frac{2}{3}\times \frac{7}{2}+\frac{1}{3}\times h \right)=\frac{1001}{6}\]

\[\frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\left( \frac{7+h}{3} \right)=\frac{1001}{6}\]

\[7+h=\frac{1001\times 7\times 2\times 2\times 3}{6\times 7\times 7\times 22}\]

\[h=\frac{1001}{77}-7\]

\[h=6\,cm\].

Area of hemispherical part of the toy \[=2\pi {{r}^{2}}\]

\[=2\times \frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\]

\[=77\,\,c{{m}^{2}}\]

\[\therefore \] The cost of painting the hemispherical part of the toy

\[=Rs.\,(77\times 10)\]

\[=Rs.\,770\]