Solution:

Comparison of Material Used in Two Cuboidal Boxes

Consider two cuboidal boxes with dimensions as follows:

• Box 1: Length = $l_1$, Width = $w_1$, Height = $h_1$
• Box 2: Length = $l_2$, Width = $w_2$, Height = $h_2$

The surface area of a cuboidal box is given by the formula:

Surface Area = $2(lw + lh + wh)$

For Box 1, the surface area is:

Surface Area$_1$ = $2(l_1w_1 + l_1h_1 + w_1h_1)$

For Box 2, the surface area is:

Surface Area$_2$ = $2(l_2w_2 + l_2h_2 + w_2h_2)$

Compare the two surface areas to determine which box requires the lesser amount of material.

Solution:

Solution:

We are given the dimensions of the suitcase: 80 cm × 48 cm × 24 cm. The suitcase needs to be covered with a tarpaulin cloth. We need to find how much tarpaulin is required to cover 100 such suitcases, given that the width of the tarpaulin is 96 cm.

The surface area of the suitcase can be calculated using the formula for the surface area of a cuboid:

Surface Area of a cuboid = $2 \times (l \times b + b \times h + h \times l)$

Here, $l = 80$ cm, $b = 48$ cm, and $h = 24$ cm.

Substitute the values:

Surface Area = $2 \times (80 \times 48 + 48 \times 24 + 24 \times 80)$

Surface Area = $2 \times (3840 + 1152 + 1920)$

Surface Area = $2 \times 6812 = 13624$ cm²

Now, the surface area of 100 such suitcases will be:

Total Surface Area = $100 \times 13624 = 1362400$ cm²

We are given that the width of the tarpaulin is 96 cm. The length of tarpaulin required can be found by dividing the total surface area by the width of the tarpaulin:

Length of tarpaulin required = $ \frac{1362400}{96} = 14200$ cm

Convert this to meters: $14200$ cm = $142$ meters

Therefore, the total length of tarpaulin required is 142 meters.

Solution:

Find the side of a cube whose surface area is 600 cm$^2$

We know that the surface area of a cube is given by the formula:

Surface Area = 6 $s^2$

Where $s$ is the side of the cube.

Given the surface area is 600 cm$^2$, we can write the equation as:

6 $s^2$ = 600

Now, divide both sides by 6:

$s^2$ = 100

Taking the square root of both sides:

$s$ = $\sqrt{100}$

$s$ = 10 cm

Solution:

Surface Area of the Cabinet

The dimensions of the cabinet are given as 1 m × 2 m × 1.5 m.

To find the surface area covered by Rukhsar, we need to calculate the area of all the surfaces except the bottom.

The surface area of a rectangular box is calculated using the formula:

Surface Area = 2 × (Length × Width + Length × Height + Width × Height)

Here, the length = 2 m, width = 1 m, and height = 1.5 m.

Step 1: Calculate the area of all six sides of the cabinet.

Surface Area = 2 × (2 × 1 + 2 × 1.5 + 1 × 1.5)

Surface Area = 2 × (2 + 3 + 1.5)

Surface Area = 2 × 6.5 = 13 m²

Step 2: Subtract the area of the bottom from the total surface area.

The bottom area is the rectangle formed by the length and width of the cabinet.

Area of the bottom = Length × Width = 2 × 1 = 2 m²

Step 3: Subtract the bottom area from the total surface area.

Surface Area covered = Total Surface Area - Area of the bottom

Surface Area covered = 13 - 2 = 11 m²

Thus, the surface area covered by Rukhsar is 11 m².

Solution:

Solution:

The cuboidal hall has dimensions:

• Length = 15 m
• Breadth = 10 m
• Height = 7 m

To calculate the total surface area to be painted, we need to find the areas of the walls and the ceiling. The surface area of the walls can be calculated using the formula for the surface area of a cuboid:

Surface area of the walls = 2 × (length × height) + 2 × (breadth × height)

Substituting the values:

Surface area of the walls = 2 × (15 × 7) + 2 × (10 × 7) = 2 × 105 + 2 × 70 = 210 + 140 = 350 m$^2$

Next, we calculate the area of the ceiling:

Area of the ceiling = length × breadth = 15 × 10 = 150 m$^2$

Now, we calculate the total area to be painted:

Total area = Surface area of walls + Area of ceiling = 350 m$^2$ + 150 m$^2$ = 500 m$^2$

Since each can of paint covers 100 m$^2$, the number of cans required is:

Number of cans = Total area / Area covered by one can = 500 / 100 = 5 cans

Therefore, Daniel will need 5 cans of paint to paint the room.

Solution:

Comparison of the Two Boxes

We are given two boxes, and we need to compare their shapes and sizes. Let us break down their similarities and differences.

Similarities

• Both boxes are rectangular prisms (cuboids), meaning they have 6 faces, 12 edges, and 8 vertices.
• Each box has 2 pairs of opposite faces that are identical in size.

Differences

• The dimensions (length, width, and height) of the two boxes may differ.
• One box may be longer, shorter, taller, or wider than the other, which will affect the lateral surface area.

Lateral Surface Area

The lateral surface area (LSA) of a rectangular prism is given by the formula:

$LSA = 2 \times (l + b) \times h$,

where:

• $l$ = length of the box
• $b$ = breadth of the box
• $h$ = height of the box

To determine which box has the larger lateral surface area, we need to substitute the dimensions of the two boxes into this formula and compare the results.

Solution:

Question:

A closed cylindrical tank of radius 7 m and height 3 m is made from a sheet of metal. How much sheet of metal is required?

Solution:

The surface area of a closed cylindrical tank consists of two parts:

• Curved surface area (CSA)
• Area of two circular ends (two bases)

1. Curved Surface Area (CSA):

The formula for the curved surface area of a cylinder is:

Curved Surface Area = $2 \pi r h$

where $r$ is the radius and $h$ is the height.

Substituting the given values, $r = 7$ m and $h = 3$ m:

CSA = $2 \pi \times 7 \times 3 = 42 \pi$ m²

2. Area of two circular ends:

The formula for the area of one circular end is:

Area of one circular end = $\pi r^2$

Thus, the area of two circular ends = $2 \pi r^2$

Substituting the value of $r = 7$ m:

Area of two circular ends = $2 \pi \times 7^2 = 2 \pi \times 49 = 98 \pi$ m²

3. Total Surface Area:

Total Surface Area = CSA + Area of two circular ends

Total Surface Area = $42 \pi + 98 \pi = 140 \pi$ m²

4. Final Answer:

Total Surface Area = $140 \pi$ m²

Approximate value of $\pi = 3.14$

Total Surface Area = $140 \times 3.14 = 439.6$ m²

Thus, the required sheet of metal is approximately 439.6 m².

Solution:

Given:

The lateral surface area of the hollow cylinder = 4224 cm²

The width of the rectangular sheet = 33 cm

The lateral surface area of a hollow cylinder is given by the formula:

Lateral Surface Area = 2$\pi$rh, where r is the radius and h is the height of the cylinder.

Step 1: Find the height of the cylinder

The rectangular sheet formed by cutting the cylinder has a width of 33 cm, which is equal to the height of the cylinder, so:

h = 33 cm

Step 2: Use the formula for lateral surface area to find the radius

We are given that the lateral surface area = 4224 cm², so:

4224 = 2$\pi$r(33)

4224 = 66$\pi$r

r = 4224 / (66$\pi$)

r = 4224 / (66 × 3.14)

r = 4224 / 207.24

r ≈ 20.37 cm

Step 3: Find the perimeter of the rectangular sheet

The length of the rectangular sheet is equal to the circumference of the cylinder, which is given by the formula:

Circumference = 2$\pi$r

Circumference = 2 × 3.14 × 20.37 ≈ 128.35 cm

The perimeter of the rectangular sheet is given by:

Perimeter = 2 × (Length + Width)

Perimeter = 2 × (128.35 + 33)

Perimeter = 2 × 161.35

Perimeter ≈ 322.7 cm

Solution:

Given:

Diameter of the road roller = 84 cm

Radius of the road roller = $\frac{84}{2} = 42$ cm

Length of the road roller = 1 m = 100 cm

Number of revolutions = 750

To find:

Area of the road leveled by the road roller

Solution:

The surface area covered by the road roller in one revolution is the area of the lateral surface of the cylinder formed by the road roller.

Formula for the lateral surface area of a cylinder: $A = 2 \pi r h$

Where: $r$ = radius of the roller = 42 cm and $h$ = height (length) of the roller = 100 cm

Substitute the values: $A = 2 \pi \times 42 \times 100 = 8400 \pi$ cm²

Since the road roller makes 750 revolutions, the total area covered is:

Total Area = $750 \times 8400 \pi = 6300000 \pi$ cm²

To convert it to square meters, we divide by $100^2 = 10000$:

Total Area in m² = $\frac{6300000 \pi}{10000} = 630 \pi$ m²

Using $\pi \approx 3.14$, we get:

Total Area ≈ $630 \times 3.14 = 1978.2$ m²

Solution:

Solution:

Given:

• Diameter of the base = 14 cm
• Radius of the base = $ \frac{14}{2} = 7 $ cm
• Height of the container = 20 cm
• Label is placed 2 cm from the top and 2 cm from the bottom, so the height of the label = $ 20 - 2 - 2 = 16 $ cm

The area of the label is the lateral surface area of the cylindrical portion that the label covers. The formula for the lateral surface area of a cylinder is:

$ A = 2 \pi r h $

Substituting the given values:

$ A = 2 \pi (7) (16) = 224 \pi $ cm²

Therefore, the area of the label is $ 224 \pi $ cm².