In Geometry, a Cube is a solid three-dimensional figure, which has 6 square faces, 8 vertices and 12 edges.
Volume of cube = a^{3}
Lateral surface area = 4a^{2}
Total surface area = 6a^{2}
To find side length of cube from the diagonal, we use the formula
Side length = a√3
Problem 1 :
Three cubes are joined end to end forming a cuboid. If side of a cube is 2 cm, find the dimensions of the cuboid thus obtained.
Solution :
Given side of a cube = 2 cm
After joining 3 cubes,
length of cuboid = 2 × 3 = 6 cm,
Height = 2 cm and Breadth = 2 cm
Dimension of the cuboid is 6 cm x 2 cm x 2 cm.
Problem 2 :
Find the lateral surface area of a cube, if its diagonal is √6 cm.
Solution :
Given, diagonal of the cube = √6 cm
Diagonal of the cube = √3 a
√3 a = √6
a = √6/√3
a = √2 cm
Lateral surface area of cube = 4a²
= 4 × (√2)²
= 4 × 2
= 8 cm²
Problem 3 :
Three cubes of metal whose edges are in the ratio 3 : 4 : 5 are melted down into a single cube whose diagonal is 12√3 cm. find the edges of the three cubes.
Solution :
Diagonal of the single cube = 12√3 cm
√3 a = 12√3
a = 12 cm
Volume of the single cube = sum of the volumes of the metallic cubes
a³ = (3x)³ + (4x)³ + (5x)³
(12)³ = 27x³ + 64x³ + 125x³
1738 = 216x³
x³ = 1728/216
x³ = 8
x³ = 2³
x = 2
Now, the edge of the first cube = 3(2) = 6 cm
Edge of the second cube = 4(2) = 8 cm
Edge of the third cube = 5(2) = 10 cm
Therefore, the edges of the three cubes are 6 cm, 8 cm, and 10 cm.
Problem 4 :
Volume of a cube is 5832 m³. Find the cost of painting its total surface area at the rate of $3.50 per m².
Solution :
Volume of a cube is 5832 m³
a³ = 5832 m³
a = 5832
a = 18 m
Total surface area = 6 × a²
= 6 × 324 = 1944 m²
Cost of painting at 3.50 per m² = 1944 × 3.50
= 6804
Hence, the cost of painting is $6804
Problem 5 :
The cube has a surface area of 216 dm². Calculate:
a) The area of one wall,
b) Edge length,
c) Cube volume.
Solution :
The cube has a surface area of 216 dm²
a) 6a^{2} = 216 a^{2} = 36 a = 6 (b) Edge length = 6 |
(c) Volume = a^{3} = 6^{3} = 216 |
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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