A cubical box has each edge 12 cm and another cuboidal box is 15 cm long, 10 cm wide and 9 cm high.
(a) Which box has the greater lateral surface area and by how much?
Answers
Answer:
Given: The length of the edge of the cubical box is 10cm and the length, breadth, and height of the cuboidal box are 12.5 cm, 10 cm, and 8 cm respectively.
A cube is a cuboid whose length, breadth, and height and equal. A cuboid has six faces and the total surface area is the sum of the surface area of the 6 faces and the Lateral surface area is the sum of the area of the four faces.
Total surface area of cube = 6a2 (where, 'a' is the side of the cube)
Total surface area of cuboid = 2(lb + bh + hl)
Lateral surface area of a cube = 4a2
Lateral surface area of cuboid = 2(l + b)h
Edge length of the cube, a = 10 cm
Length of the cuboid, l = 12.5 cm
Breadth of the cuboid, b = 10 cm
Height of the cuboid, h = 8 cm
Lateral surface area of the cube = 4a2
= 4 × (10 cm)2
= 4 × 100 cm2
= 400 cm2
Lateral surface area of the cuboid = (l + b)h
= 2 × (12.5 cm + 10 cm) × 8 cm
= 2 × 22.5 cm × 8 cm
= 360 cm2
We see that, the cubical box has a greater lateral surface area by (400 cm2 - 360 cm2) = 40 cm2
Total surface area of the cube = 6a2
= 6 × (10 cm)2
= 6 × 100 cm2
= 600 cm2
Total surface area of the cuboid = 2(lb + bh + hl)
2(lb + bh + hl) = 2 × (12.5 cm × 10 cm + 10 cm × 8 cm + 8 cm × 12.5 cm)
= 2 × (125 cm2 + 80 cm2 + 100 cm2)
= 2 × 305 cm2
= 610 cm2
Cubical box has a smaller total surface area by (610 cm2 - 600 cm2) = 10 cm2
Thus, the cubical box has a greater lateral surface area by 40 cm2 and the cubical box has a smaller total surface area by 10 cm2.