From a rectangular solid of metal 42 cm by 30 cm by 20 cm, a conical cavity of diameter 14 cm and depth 24 cm is drilled out. Find: (i) the surface area of the remaining solid (ii)the volume of remaining solid
From a rectangular solid of metal 42 cm by 30 cm by 20 cm, a conical cavity of diameter 14 cm and depth 24 cm is drilled out. Find: (i) the surface area of the remaining solid (ii)the volume of remaining solid

Selina Solutions Concise Class 10 Maths Chapter 20 ex. 20(E) - 2

Dimensions of rectangular solid l = 42 cm, b = 30 cm and h = 20 cm

Conical cavity’s diameter = 14 cm

its radius = 7 cm

Depth (height) = 24 cm

(i) Total surface area of cuboid

    \[\begin{array}{*{35}{l}} =\text{ }2\text{ }\left( lb\text{ }+\text{ }bh\text{ }+\text{ }lh \right)  \\ =\text{ }2\text{ }\left( 42\text{ x }30\text{ }+\text{ }30\text{ x }20\text{ }+\text{ }20\text{ x }42 \right)  \\ =\text{ }2\text{ }\left( 1260\text{ }+\text{ }600\text{ }+\text{ }840 \right)  \\ =\text{ }2\text{ }\left( 2700 \right)  \\ =\text{ }5400\text{ }c{{m}^{2}}  \\ \end{array}\]

Diameter of the cone = 14 cm

its radius = 14/2 = 7 cm

Area of circular base

    \[\begin{array}{*{35}{l}} =\text{ }\pi {{r}^{2}}~=\text{ }22/7\text{ x }7\text{ x }7\text{ }=\text{ }154\text{ }c{{m}^{2}}  \\ l\text{ }=\text{ }{{\left( {{7}^{2}}~+\text{ }{{24}^{2}} \right)}^{1/2}}~=\text{ }{{\left( 49\text{ }+\text{ }576 \right)}^{1/2}}~=\text{ }{{\left( 625 \right)}^{1/2}}~=\text{ }25  \\ Area\text{ }of\text{ }curved\text{ }surface\text{ }area\text{ }of\text{ }cone\text{ }=\text{ }\pi rl\text{ }=\text{ }22/7\text{ x }7\text{ x }25\text{ }=\text{ }22\text{ x }25\text{ }=\text{ }550\text{ }c{{m}^{2}}  \\ Surface\text{ }area\text{ }of\text{ }remaining\text{ }part\text{ }=\text{ }5400\text{ }+\text{ }550\text{ }\text{ }154\text{ }=\text{ }5796\text{ }c{{m}^{2}}  \\ \end{array}\]

(ii) Volume of the rectangular solid = (42 x 30 x 20) cm3 = 25200 cm3

Radius of conical cavity (r) = 7 cm

Height (h) = 24 cm

    \[\begin{array}{*{35}{l}} Volume\text{ }of\text{ }cone\text{ }=\text{ }1/3\text{ }\pi {{r}^{2}}~h  \\ =\text{ }1/3\text{ x }22/7\text{ x }7\text{ x }7\text{ x }24  \\ =\text{ }1232\text{ }c{{m}^{3}}  \\ \end{array}\]