The perimeter of the triangular base of a right prism is 15 cm and radius of the in circle of the triangular base is 3 cm. If the volume of the prism be 270 cm³, then the height of the prism is

Solution (By Examveda Team)

$$\eqalign{ & r{\text{ - inradius of incircle of triangle}} \cr & {\text{Perimeter}} = 15{\text{ cm }}\left( {{\text{given}}} \right) \cr & \therefore {\text{Semiperimeter}}\left( S \right) = \frac{{15}}{2}{\text{cm}} \cr & {\text{Inradius of any triangle}} \cr & r \Rightarrow \frac{\Delta }{S} \cr & r = \frac{{{\text{area}}}}{{{\text{semiperimeter}}}} \cr & {\text{Where }}\Delta {\text{ is the area of triangle }} \cr & \therefore r{\text{ }} = {\text{ }}3{\text{ cm }}\left( {{\text{given}}} \right) \cr & \Rightarrow 3 = \frac{{{\text{area of triangle}}}}{{\frac{{15}}{2}}} \cr & \Rightarrow 3 \times \frac{{15}}{2} = {\text{area of triangle}} \cr & \Rightarrow \frac{{45}}{2}{\text{cm}} = {\text{area of triangle}} \cr & \therefore {\text{Volume of prism}} \cr & \Rightarrow 270{\text{ c}}{{\text{m}}^3}\,\left( {{\text{given}}} \right) \cr & \therefore 270 = h \times \frac{{45}}{2} \cr & h = 12{\text{ cm}} \cr} $$