Mensuration 3D - Study Mode

[#46] A solid cube has side 8 cm. It is cut along diagonals of top face to get 4 equal parts. What is the total surface (in cm 2 ) of each part?
Correct Answer

(A) 96 + 64√2

Explanation

Solution: Total surface area of each part is = 2 × Area of base + Perimeter of base × Height $$eqalign{
& = 2 imes frac{1}{2} imes 4sqrt 2 imes 4sqrt 2 + left( {8 + 8sqrt 2 }
ight) imes 8 cr
& = 32 + 64 + 64sqrt 2 cr
& = 96 + 64sqrt 2 cr} $$

[#47] The base of a right pyramid is a square of side 8√2 cm and each of its slant edge is of length 10 cm. What is the volume (in cm 3 ) of the pyramid?
Correct Answer

(A) 256

Explanation

Solution: Length of each side = 8√2 cm Diagonal = √2a = √2 × 8√2 = 16 OC $$ = frac{{ ext{d}}}{2} = frac{{16}}{2} = 8$$ Slant edge = EC = 10 cm In ΔOEC EC 2 = OE 2 + OC 2 10 2 = OE 2 + 8 2 OE 2 = 6 2 Height = OE = 6 cm Volume of the pyramid = $$frac{1}{3}$$ × Area of base × Height = $$frac{1}{3}$$ × (8√2) 2 × 6 = 256

[#48] The base of a solid right prism of height 10 cm is a square and its volume is 160 m 3 . What is the total surface area of the prism (in cm 2 )?
Correct Answer

(C) 192

Explanation

Solution: 10 cm ⇒ height Volume of Prism = Base Area × height 160 = a 2 × 10 a = 4 Total surface area = 2Base Area + Base per meter × length = 2 × 16 + 16 × 10 = 32 + 160 = 192

[#49] A solid lead sphere of radius 11 cm is melted and recast into small solid spheres of radius 2 cm each. How many maximum number (in integer) of such spheres can be made?
Correct Answer

(B) 166

Explanation

Solution: $$eqalign{
& {R^3} = n imes {r^3} cr
& 11 imes 11 imes 11 = n imes {2^3} cr
& frac{{1331}}{8} = n cr
& n = 166 cr
& n = 166{ ext{ }}left( {{ ext{integer}}}
ight) cr} $$

[#50] A hemisphere is kept on top of a cube. Its front view is shown in the given figure. The total height of the figure is 21 cm. The ratio of curved surfaces area of hemisphere and total surface area of cube is 11 : 42. What is the total volume (in cm 3 ) of figure?
Correct Answer

(B) 3462.67

Explanation

Solution: $$frac{{{ ext{Curved surfaces area of hemisphere}}}}{{{ ext{Total surface area of cube}}}} = frac{{2pi {r^2}}}{{6{a^2}}}$$ 3r = 21 r = 7 cm Total volume of figure = Volume of hemisphere + Volume of cube = $$frac{2}{3}$$πr 3 + a 3 = $$frac{2}{3}$$ × $$frac{{22}}{7}$$ × 7 × 7 × 7 + 14 3 = 718.66 + 2744 = 3462.67 cm 3