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

Answer: Option D

Solution

Answer: Option D

Solution:

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