Question

A toy is made in the form of hemisphere surmounted by a right cone whose circular base coincides with the plane surface of hemisphere. The radius of the base of the cone is $3\cdot 5meters$ and its volume is two-third of the hemisphere. Calculate the height of the cone and the surface area of the toy, correct to two places of decimal.

Answer 1

Radius of core $=3.5m$

radius of hemisphere $=3.5m$

$\frac{volumeofcore}{volumeofhemisphere}=\frac{2}{3}$

$\frac{\frac{\pi r^{2}h}{3}}{\frac{2}{3}\pi r^3}=\frac{2}{3}$

$\frac{h}{2r}=\frac{2}{3}$

$h=\frac{4}{3}r=\frac{4}{3}\times 3.6=4.66m$

Surface of toy $=\pi r\sqrt{h^2+r^2}+2\pi r^2$

$=\pi r(\sqrt{h^2+r^2}+2r)$

$=\frac{22}{7}\times 3.5(\sqrt{(3.5{)}^2+(4.66{)}^2}+2\times 3.5)$

$=\frac{22}{7}\times 3.5(5.83+7)$

$=141.13m^2$