A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long and 10 cm wide and 8 cm high.

• (i) Which box has the greater lateral surface area and by how much?
• (ii) Which box has the smaller total surface area and by how much?

To compare the lateral and total surface areas of the two boxes, we first need to calculate these areas for each box.

Lateral Surface Area Calculation

Cubical Box

The lateral surface area of a cube is calculated using the formula:

Lateral Surface Area = 4 × edge²

For the cubical box with an edge of 10 cm:

Lateral Surface Area = 4 × (10 cm)² = 4 × 100 cm² = 400 cm²

Cuboidal Box

The lateral surface area of a cuboid is calculated using the formula:

Lateral Surface Area = 2 × (length + width) × height

For the cuboidal box (12.5 cm long, 10 cm wide, and 8 cm high):

Lateral Surface Area = 2 × (12.5 cm + 10 cm) × 8 cm = 2 × 22.5 cm × 8 cm = 360 cm²

Comparison of Lateral Surface Areas

The cubical box has a lateral surface area of 400 cm², while the cuboidal box has 360 cm². Thus:

The cubical box has the greater lateral surface area by 40 cm².

Total Surface Area Calculation

Cubical Box

The total surface area of a cube is calculated using the formula:

Total Surface Area = 6 × edge²

For the cubical box:

Total Surface Area = 6 × (10 cm)² = 6 × 100 cm² = 600 cm²

Cuboidal Box

The total surface area of a cuboid is calculated using the formula:

Total Surface Area = 2 × (length × width + width × height + height × length)

For the cuboidal box:

Total Surface Area = 2 × (12.5 cm × 10 cm + 10 cm × 8 cm + 8 cm × 12.5 cm)

Total Surface Area = 2 × (125 cm² + 80 cm² + 100 cm²) = 2 × 305 cm² = 610 cm²

Comparison of Total Surface Areas

The cubical box has a total surface area of 600 cm², while the cuboidal box has 610 cm². Therefore:

The cubical box has the smaller total surface area by 10 cm².