• # question_answer A tool is made up of a cone on top of a cylinder as shown in the figure below. The cylinder has a height of 15 cm and a radius of 5 cm. The volume of the cone is $100\,\pi \,c{{m}^{3}}$. If O is the vertex of the cone and AB is the diameter of the base of the cone and C is its centre. Points O, A, B and C are in the same plane. Calculate the lateral surface area of the tool. A) $675.714\,c{{m}^{2}}$   B)       $685.4\,c{{m}^{2}}$C) $695.4\,c{{m}^{2}}$         D)        $775.4\,c{{m}^{2}}$E) None of these

Explanation Option [a] is correct. Note that the radius of the cylinder and the radius of the base of the cone have the same size. We first use the formula of the volume of the cone to find its height H: $\left( \frac{1}{3} \right)\pi \,5\,\,{{\times }^{2}}h=100\,\,\pi \Rightarrow h=12\,\,cm$ ${{\ell }^{2}}+{{h}^{2}}+{{r}^{2}}\Rightarrow \ell =13$ The area of lateral surface of the cone and that of the cylinder is added to obtain the total area of the surface. So, lateral surface area of the figure = $2\pi rh+\pi r\ell =\pi r\left( 2h+\ell \right)$ = $\frac{22}{7}\times 5\left( 30+13 \right)=675.714\,c{{m}^{2}}$