Three cubes with sides in the ratio 1 : 6 : 8 are melted to form a single cube whose diagonal is 27 cm. the sides of the cubes are:
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Solution
Let the sides of the three cubes be 1x, 6x and 8x.
Then, Volume of the new cube = [(1x)3 + (6x)3 + (8x)3] = 729x3.
Side of the new cube = (729x3)1/3 = 9x.
Diagonal of new cube = 9x cm.
⇒ 9
⇒ x = 3
∴ Side of cubs are = 3 × 1, 3 × 6, 3 × 8 = 3 cm, 18 cm, 24 cm.
If each edge of a cube is doubled, then its volume will be :
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Solution
Let original side is a cm
So, volume = a3 cm3
New side = 2a
So, new volume = 8a3 cm3
A cube of edge 15 cm is cut into small cubes of each edge 3 cm. The ratio of the total surface area of one of the small cubes to that of the large cube, what is the ratio of ?
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Solution
Required ratio = (6 × 3 × 3) : (6 × 15 × 15) = 1 : 25
A cistern 8 m long and 6 m wide contains water up to a height of 1 m 75 cm. Find the total area of the wet surface.
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Solution
Area of the wet surface = 2[lb + bh + hl] – lb
= 2 [bh + hl] + lb
= 2[(6 × 1.75 + 8 × 1.75)] + 6 × 8= 97 square meter
The size of a wooden block is 6 cm × 15 cm × 24 cm. How many such blocks will be required to construct a solid wooden cube of minimum size?
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Solution
Side of smallest cube = L.C.M of 6, 15, 24 = 120 cm
Volume of the cube = (120 × 120 × 120) cm3 = 1728000 cm3
Volume of the block = (6 × 15 × 24) cm3 = 2160 cm3
Required Number of blocks = = 800
A metallic hemisphere is melted and recast into the shape of a cone with the same base radius R as that of the hemisphere. If H is the height of the cone, then :
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Solution
∵ Volume of Hemisphere =
Volume of cone =
According to question,
⇒
If a metallic cuboid weighs 6.75 kg, how much would a small cuboid of metal weigh, if all dimensions are reduced to one-third of the original.
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Solution
Let the dimensions of the larger cuboid be x, y and z.
Then, volume of the larger cuboid = xyz.
Volume of the smaller cuboid =
∴ Weight of the smaller cuboid = kg = 0.25 kg.
If the surface area of a cube is 864 cm2, then what will be length of its diagonal ?
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Solution
∵ Surface area of a cube = 6 × (side)2
⇒ 6 × (side)2 = 864
⇒ (side)2 = 144
⇒ side = 12 cm.
∴ Diagonal = cm.
60 men took a dip in water tank 60 m long and 25 m broad on a religious day. If the average displacements of water by a man is 5 m3, then what will be rise of the water level in water tank ?
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Solution
Total volume of water displaced = (5 × 60) m3 = 300 m3
∴ Rise in water level = m
= 20 cm.
Find the surface area of a brick whose sides are 12 cm × 6 cm × 5 cm.
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Solution
Surface area of a cuboid = 2(lb + bh + hl) cm square.
So, Surface area of a brick = 2(12 × 6 + 6 × 5 + 12 × 5) cm square
= 2 × 162 cm square = 324 cm square