MENSURATION-IV

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Joshua Griffith
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1 MENSURATION-IV Theory: A solid is figure bounded by one or more surfce. Hence solid hs length, bredth nd height. The plne surfces tht bind solid re clled its fces. The fundmentl difference between plne figure nd solid figure is tht the plne figure lies in plne nd solid figure lies in spce. There re two types of three-dimensionl figures (1) The solid figure in which ny of the cross section is the sme throughout. E.g. Cube, Cuboid, Cylinder etc. () The solid figure in which none of the cross-sections is sme throughout. E.g. Cone, Sphere, Pyrmid etc. CUBOID: A cuboid is bounded by 6 rectngulr fces. The opposite fces of rectngulr solid re equl rectngles lying in prllel plnes. E F A B h b G H C D A The res of three different fces be A1, A nd A then A1 = lb A = bh = lh Surfce re = ( A 1 + A + A ) = (lb + bh + lh) Volume = Are of ny fce corresponding height V = lb h = lbh Digonl (d) = l + b + h Digonl is the biggest possible dimension of cuboid. Also A 1 A A = ( lb) ( bh) ( lh) = ( lbh) = V V = A 1A A CUBE: A Cube is bounded by six squre fces i.e. if the length,bredth nd height of cuboid re ll equl then it is clled cube. If ech side of the cube is of units, then its surfce re(s.a) =6 nd Its Volume(V) = Digonl of cube will be d =

2 PROBLEMS 1. Ech edge of cube is decresed by 0%. The percentge of decrese in the surfce re of the cube is 1) 44% ) 6% ) 0% 4) 60% 5) None of ANSWER: Edge of the cube be 5 then its surfce re = 6 5 = After reduction new edge of the cube = 5 = 5/ = 4 5/ New surfce re of the cube = 6 4 = Surfce re reduces by 100 = 100 = 6% Shortcut method: (i) If ech edge of cube incresed by x % then the surfce re increses by x S = x + % (ii) If ech edge of cube decresed by x % then the surfce re decreses by x S = x % 0 In the bove problem x = 0%, then S = 0 % = (40 4)% = 6%. A cuboid ( cm 4 cm 5 cm) is cut into unit cubes. The rtio of the totl surfce re of ll the unit cubes to tht of the cuboid is 1) 180 : ) 180 : 9 ) 180 : 6 4) 180 : 47 5) None of The dimensions of cuboid re 4 5 Its surfce re (S.A) = ( ) = 94 cm If the cuboid is cut into unit cubes, then the number of unit cubes so formed = 4 5 = 60 But surfce re of ech unit cube = 6 1 = 6 Totl surfce re of unit cubes = 6 60 = 60 Required rtio = 60 : 94 = 180 : 47. If the digonl of cube is 10 cm, then its surfce re will be 1) 500 cm ) 550 cm ) 600 cm 4) 650 cm 5) None of ANSWER: Digonl (d) of cube = 10 Its side = d 10 = = 10 Surfce re (S.A) of cube = 6 = 6 10 = 600 cm

3 4. If the volume of cube is 16 cm, then the surfce re of the cube will be 1) 14 cm ) 16 cm ) 18 cm 4) 0 cm 5) None of ANSWER: Volume (V) of cube = = 16 = 16 = 6 Its surfce re = 6 = 6 6 = 16 cm 5. If six cubes, ech of 10 cm edge, re joined end to end, then the surfce re of the resulting solid will be 1) 600 cm ) 000 cm ) 600 cm 4) 400 cm 5) None of ANSWER: When six cubes re joined end to end, cuboid will be formed whose length is 6 10 = 60 cm, bredth 10 cm nd height 10 cm respectively i.e. l = 60, b = 10 & h = 10 Surfce re of cuboid = ( ) = ( ) = 600 sq cm 6. If three cubes of copper, ech with n edge of 6 cm, 8 cm nd 10 cm respectively re melted to form single cube, then the digonl of the new cube will be 1) 18 cm ) 19 cm ) 19.5 cm 4) 0.8 cm 5) None of If three cubes re melted to form single lrger cube then the volume of lrger cube so formed will be equl to the sum of the volumes of the three cubes. Volume of the lrger cube = = = 178 Side of lrger cube = 178 = 1 Digonl of lrger cube = 1 = = 0.8 cm 7. A swimming pool 9 m wide nd 1 m long is 1 m deep on the shllow side nd 4 m deep on the deeper side. Its volume is 1) 408 m ) 60 m ) 70 m 4) 08 m 5) None of ANSWER: The cross-section of the swimming pool is trpezium whose prllel sides re 1 m nd 4 m nd hving perpendiculr distnce of 9 m Are of cross-section = 9 =.5 sq m Volume of swimming pool =.5 1 = 70 cu.m 8. The length, bredth nd height of cuboid re in the rtio 1 : :. The length, bredth nd height of the cuboid re incresed by 100%, 00% nd 00% respectively. Then the increse in the volume of the cuboid is 1) 5 times ) 6 times ) 1 times 4) 17 times 5) None of Length, bredth nd height of cuboid be x, x nd x respectively, then its volume = x x x = 6x

4 When length, bredth nd height re incresed by 100%, 00% nd 00% respectively, New length = x = x New bredth = x = 6x New height = x = 9x New volume = x 6x 9x = 108x 108x 6x 10x Increse in volume = = 6x = 17 times 6x 9. A cube of led with edges mesuring 6 cm ech is melted nd formed into 7 equl cubes. Wht will be the length of the edges of the new cubes? 1) cm ) 4 cm ) cm 4) 1 cm 5) None of ANSWER: The edge of ech smller cube be. Then totl volume of 7 cubes = 7 But totl volume of 7 cubes is equl to volume of cube of edge 6 cm 7 = 6 = = = 8 7 = 10. The edges of cuboid re in the rtio 1 : : nd its surfce re is 88 cm. The volume of the cuboid is 1) 10 cm ) 64 cm ) 48 cm 4) 4 cm 5) None of ANSWER: The edges of cuboid re in the rtio of 1 : :. So the edges cn be ssumed s x, x nd x Surfce re (S.A) = ( x x + x x + x x) = (11x ) = x x = 88 x = The dimensions of cuboid will be, 4 nd 6. The volume of cuboid = 4 6 = 48 cm. 11. The res of three djcent fces of cuboid re,b nd c. If the volume of the cuboid is V, then V is equl to 1) bc ) (b + bc + c) c ) b 4) ( + b + c) 5) None of ANSWER: 1 If the three djcent dimensions re x, y nd z, then x y = y z = b x z = c x y = yz = b x z = c ( x y) (yz) (z x ) = bc

5 x y z = bc But x y z = V V = bc 1. The sum of the length, bredth nd depth cuboid is 19 cm nd its digonl is 5 5. Its surfce re is 1) 61 cm ) 15 cm ) 6 cm 4) 144 cm 5) None of ANSWER: If l, b nd h re the three dimensions of cuboid then l + b + h = 19, l + b + h = 5 5 ( l + b + h) = l + b + h + ( lb + bh + hl) 19 = ( ) (l b + bh + hl ) ( l b + bh + l h) = = 6 cm Surfce re = 6 cm 1. Wht is the time needed to empty cubodil wter reservoir 10 m long, 9 m wide nd m deep t the rte of 45 L/m? 1) 100 h ) 90 h ) 80 h 4) 60 h 5) None of lit] ANSWER: 1 Volume of wter in reservoir = 10 9 = 70 cu. m = lit = lit [ 1 cu. m = Time to empty tnk = = 6000 min = = 100 h 45 60

Naming 3D objects. 1 Name the 3D objects labelled in these models. Use the word bank to help you.

Naming 3D objects. 1 Name the 3D objects labelled in these models. Use the word bank to help you. Nming 3D ojects 1 Nme the 3D ojects lelled in these models. Use the word nk to help you. Word nk cue prism sphere cone cylinder pyrmid D A C F A B C D cone cylinder cue cylinder E B E prism F cue G G pyrmid