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$\frac{5}{6}c{m}^3$.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 $c{m}^2$.$(Take\pi =22/7)$

Solution:

Given that,

Radius of the hemisphere = 3.5 cm

Volume of the solid wooden toy = 166$\frac{5}{6}c{m}^3$

As, Volume of the solid wooden toy = Volume of the cone + Volume of the hemisphere = 166$\frac{5}{6}c{m}^3$

$\frac{1}{3}\times \pi r^{2}L+\frac{2}{3}\times \pi r^3=\frac{1001}{6}$

$\frac{1}{3}\times \pi {3.5}^{2}L+\frac{2}{3}\times \pi {3.5}^3=\frac{1001}{6}$

L + 7 = 13

L = 6 cm

Height of the solid wooden toy = Height of the cone + Radius of the hemisphere

= 6 + 3.5

= 9.5 cm

Now, curved surface area of the hemisphere = $2\pi R^2=2\pi \times (3.5{)}^2$ = 77 cm²

Cost of painting the hemispherical part of the toy = 10 77 = Rs. 770