The surface area of a cuboid formed by joining two cubes of side a face to face is __________
10a2
Explanation:
Let “a” be the side of two cubes.
When the two cubes are joined face to face, the figure obtained should be a cuboid having the same breadth and height. As the combined cube has a length twice of the length of a cube.
It means that l = 2a, b = a and h = a
Hence, the total surface area of cuboid = 2(lb + bh + hl)
= 2(2a × a + a × a + a × 2a)
Simplify the above expression, we get
= 2[2a2 + a2 + 2a2]
= 10a2
A cube of side 5 cm is cut into 1 cm cubes. The percentage increase in volume after such cutting is __________.
No change
Explanation:
Volume of cube = 53 = 125
Now, when the cube is cut into 1 cubic cm, we will get 125 small cubes
Therefore, the volume of the big cube = volume of 125 cm with 1 cubic cm.
It means that, there is no change in the volume.
A cube of side 4 cm is painted on all its sides. If it is sliced in 1 cubic cm cubes, then number of such cubes that will have exactly two of their faces painted is __________.
If R is the radius of the base of the hat, then the total outer surface area of the hat is
(a) πr (2h + R) (b) 2πr (h + R)
(c) 2 πrh + πR2 (d) None of these
The correct answer is option (c) 2 prh + pR2
Explanation:
The total surface area of a hat = CSA + TSA + Base Surface Area
= 2prh + pr2 + p (R2-r2)
= 2prh + pR2
Ramesh has three containers.
(a) Cylindrical container A having radius r and height h,
(b) Cylindrical container B having radius 2r and height 1/2 h, and
(c) Cuboidal container C having dimensions r × r × h
The arrangement of the containers in the increasing order of their volumes is
(a) A, B, C
(b) B, C, A
(c) C, A, B
(d) cannot be arranged
The correct answer is option (c) C, A, B
Explanation:
(i) If the cylinder have radius r and height h, then the volume will be pr2h
(ii) If the cylinder have radius 2r and height (1/2)h, then the volume will be 2pr2h
(ii) The volume of the cuboidal container with dimensions is r2 h
Then, the arrangement of the containers in the increasing order of their volumes is C, A, B
The surface areas of the six faces of a rectangular solid are 16, 16, 32, 32, 72 and 72 square centimetres. The volume of the solid, in cubic centimetres, is
(a) 192 (b) 384 (c) 480 (d) 2592
The correct answer is option (a) 192
Explanation:
It is given that, the solid has a rectangular faces, hence,
lb=16 …(1)
bh = 32 ….(2)
lh = 72 …(3)
Multiply the equations (1), (2), (3), we will get
(l)2(b) 2 (h) 2 = (16)(32)(72) = 36864
lbh = 192
Therefore, the volume of a solid is 192 cubic centimetre
Two cubes have volumes in the ratio 1:64. The ratio of the area of a face of first cube to that of the other is
(a) 1:4 (b) 1:8 (c) 1:16 (d) 1:32
The correct answer is option (c) 1:16
Explanation:
Let a and b be two cubes
It is given that, a3/b3 = 1/64
Then a/b = 1/4
Thus, the ratio of the areas are:
(a/b)2 = (1/4)2 = 1/16
The ratio of radii of two cylinders is 1: 2 and heights are in the ratio 2:3. The ratio of their volumes is
(a) 1:6 (b) 1:9 (c) 1:3 (d) 2:9
The correct answer is option (a) 1:6
Explanation:
Assume that r and R be the radii of the two cylinders and h and H be the height of the two cylinders
It is given that r/R = ½ and h/H = 2/3
We know that the volume of a cylinder = pr2 h
Now, v/V = pr2 h / pR2 H
v/ V = (r/R)2 (h/H)
v/V = (1/2)2 (2/3)
v/V = (1/4) (2/3) = 1/6
Therefore, the ratio of their volume is 1/6
A covered wooden box has the inner measures as 115 cm, 75 cm and 35 cm and thickness of wood as 2.5 cm. The volume of the wood is
(a) 85,000 cm3 (b) 80,000 cm3 (c) 82,125 cm3 (d) 84,000 cm3
The correct answer is option (c) 82,125 cm3
Explanation:
The thickness of the wooden box is 2.5 cm
Then the outer measure of the wooden box be 115+5, 75+5, 35+5
Thus, the outer volume be = (120)(80)(40)
Outer volume = 384000 cm3
Given that, the inner volume = (115)(80)(40)
Inner volume = 301875 cm3
Hence, the volume of a wood = Outer volume – Inner volume
V = 384000 – 301875 cm3
V= 82125 cm3
Three cubes of metal whose edges are 6 cm, 8 cm and 10 cm respectively are melted to form a single cube. The edge of the new cube is
(a) 12 cm (b) 24 cm (c) 18 cm (d) 20 cm
The correct answer is option (a) 12 cm
Explanation:
Given that, the sum of the volume of the three metal cubes = 63 + 83 +103
V = 216+ 512+ 1000
V = 1728 cm3
Let the side of the new cube be “a”
Therefore, the volume of the new cube = sum of the volume of the three cubes
a3 = 1728
Hence, a = 12 cm
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