Q1. A metallic sheet of the rectangular shape with dimensions 50cm x 40cm. From each one of its corner, a square of 5cm is cut off. An open box is made of the remaining sheet. Find the volume of the box and the total surface area of the box.

length of base = 50-10 = 40 cm

width of box = 40-10 = 30 cm
height of box = 5 cm

volume = length x width x height
= ...... cm^3

40 x 30 x 5 = 6000 cm^3

total surface area =2(lb+bh+hl)
=2(1200+150+200)
=2x1550
=3100

To find the volume of the box, we need to determine the dimensions of the box first.

Step 1: Calculate the length and width of the remaining sheet after cutting off the squares from each corner:
Length of remaining sheet = (Original length) - 2*(Length of cut-off square)
= 50cm - 2*(5cm)
= 50cm - 10cm
= 40cm

Width of remaining sheet = (Original width) - 2*(Width of cut-off square)
= 40cm - 2*(5cm)
= 40cm - 10cm
= 30cm

Step 2: Determine the height of the box. Since the cut-off squares are bent to form the sides of the box, the height of the box will be equal to the side length of the cut-off squares, which is 5cm.

Therefore, the dimensions of the box are:
Length = 40cm
Width = 30cm
Height = 5cm

Step 3: Calculate the volume of the box:
Volume = Length * Width * Height
= 40cm * 30cm * 5cm
= 6000 cm³

The volume of the box is 6000 cm³.

To find the total surface area of the box, we need to calculate the areas of all six sides.

Step 4: Calculate the area of the bottom of the box:
Area of bottom = Length * Width
= 40cm * 30cm
= 1200 cm²

Step 5: Calculate the area of the top of the box, which is the same as the area of the bottom:
Area of top = Area of bottom
= 1200 cm²

Step 6: Calculate the area of the front and back sides of the box:
Area of front and back sides = Length * Height
= 40cm * 5cm
= 200 cm²

Step 7: Calculate the area of the two side faces of the box:
Area of side faces = Width * Height
= 30cm * 5cm
= 150 cm²

Step 8: Calculate the total surface area of the box:
Total surface area = (2 * Area of bottom) + (2 * Area of front and back sides) + (2 * Area of side faces)
= (2 * 1200 cm²) + (2 * 200 cm²) + (2 * 150 cm²)
= 2400 cm² + 400 cm² + 300 cm²
= 3100 cm²

The total surface area of the box is 3100 cm².

To find the volume and surface area of the open box, we need to determine the dimensions of the remaining sheet after the squares are cut off from each corner.

Given:
Dimensions of the rectangular sheet = 50 cm x 40 cm

Step 1: Calculate the new dimensions of the remaining sheet
When a square of 5 cm is cut off from each corner, the length and width of the remaining sheet will reduce by 5 cm each.

The length of the remaining sheet = original length - 2 * 5 cm = 50 cm - 2 * 5 cm = 40 cm
The width of the remaining sheet = original width - 2 * 5 cm = 40 cm - 2 * 5 cm = 30 cm

Step 2: Calculate the volume of the box
The volume of the box is given by the formula: Volume = Length x Width x Height

Since the height of the box is not given explicitly, we need to determine the height using the information provided.

The height of the box will be the height of the cut-off squares, which is 5 cm.

Therefore, the volume of the box = Length x Width x Height
= 40 cm x 30 cm x 5 cm
= 6000 cm³

So, the volume of the open box is 6000 cm³.

Step 3: Calculate the total surface area of the box
The surface area of the box can be calculated by summing the areas of each face.

The box has 6 faces:
1. Top face = Length x Width = 40 cm x 30 cm
2. Bottom face = Length x Width = 40 cm x 30 cm
3. Front face = Length x Height = 40 cm x 5 cm
4. Back face = Length x Height = 40 cm x 5 cm
5. Left face = Width x Height = 30 cm x 5 cm
6. Right face = Width x Height = 30 cm x 5 cm

Total surface area of the box = 2 * (Top face + Bottom face) + 2 * (Front face + Back face) + 2 * (Left face + Right face)
= 2 * (40 cm x 30 cm) + 2 * (40 cm x 5 cm) + 2 * (30 cm x 5 cm)
= 2400 cm² + 400 cm² + 300 cm²
= 3100 cm²

So, the total surface area of the open box is 3100 cm².

Therefore, the volume of the box is 6000 cm³ and the total surface area of the box is 3100 cm².