Question

A solid wooden toy is in the form of a 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 $166\frac{5}{6}$ cm³. Find the height of the toy. Also, find the cost of painting the hemispherical part of the toy at the rate of Rs 10 per cm² . (Take$\mathrm{\pi }=\frac{22}{7}$). [CBSE 2015]

Answer 1

Volume of solid wooden toy = $166\frac{5}{6}$ cm³
⇒ Volume of cone + Volume of hemisphere = $166\frac{5}{6}$
$$\begin{gathered} \Rightarrow \frac{1}{3}\mathrm{\pi }{r}^{2}h+\frac{2}{3}\mathrm{\pi }{r}^3=\frac{1001}{6} \\ \Rightarrow \frac{1}{3}\times \frac{22}{7}\times {\left(3.5\right)}^2\times h+\frac{2}{3}\times \frac{22}{7}\times {\left(3.5\right)}^3=\frac{1001}{6} \\ \Rightarrow \frac{1}{3}\times \frac{22}{7}\times {\left(3.5\right)}^2\left(h+7\right)=\frac{1001}{6} \\ \Rightarrow \frac{38.5}{3}\left(h+7\right)=\frac{1001}{6} \\ \Rightarrow h+7=13 \\ \Rightarrow h=6\mathrm{cm} \end{gathered}$$
Height of the solid wooden toy
= Height of cone + Radius of hemisphere
= 6 + 3.5
= 9.5 cm
Now, curved surface area of hemisphere = $2\mathrm{\pi }{r}^2$
$$\begin{gathered} =2\times \frac{22}{7}\times {\left(3.5\right)}^2 \\ =77{\mathrm{cm}}^2 \end{gathered}$$
Cost of painting the hemispherical part of the toy = 10 ⨯ 77 = Rs 770