All questions of Mensuration for Class 8 Exam

Because 14cm /7cm is 2cm

By the method of tsa

Understanding the Problem
To find the height of the water level in the swimming pool, we need to use the formula for the volume of a rectangular prism (which the pool resembles).
Volume Formula
The volume (V) of a rectangular prism is calculated as:
- V = length × width × height
In this case:
- Length = 260 m
- Width = 140 m
- Volume of water = 54,600 m³
Finding the Height
Rearranging the volume formula to find height (h):
- h = V / (length × width)
Now, plug in the values:
- h = 54,600 m³ / (260 m × 140 m)
Calculating the Area of the Base
First, calculate the area of the base (length × width):
- Area = 260 m × 140 m = 36,400 m²
Calculating the Height
Now, substitute the area back into the height formula:
- h = 54,600 m³ / 36,400 m²
Final Calculation
Performing the division:
- h = 54,600 / 36,400 = 1.5 m
Thus, the height of the water level in the swimming pool is:
Correct Answer
- 1.5 m (Option B)
This calculation shows that when 54,600 cubic meters of water is pumped into the pool, it raises the water level to 1.5 meters.

Given:
Circumference of the base of the cylinder = 198 cm
Height of the cylinder = 30 cm

To find:
Curved surface area of the cylinder

Solution:
First, let's find the radius of the base of the cylinder using the circumference formula:

Circumference = 2πr

Given that the circumference is 198 cm, we can equate it to 2πr:

198 = 2πr

Dividing both sides by 2π, we get:

r = 198 / (2π)

Now, let's calculate the curved surface area of the cylinder:

Curved surface area = 2πrh

Substituting the given values, we get:

Curved surface area = 2π * (198 / (2π)) * 30

Simplifying further, we have:

Curved surface area = 198 * 30

Curved surface area = 5940 cm²

Therefore, the curved surface area of the cylinder is 5940 cm², which corresponds to option B.

**Solution:**

To find the total surface area of a cylinder, we need to calculate the areas of the two circular bases and the curved surface area.

Given:
Base radius = 10.5 cm
Length = 18 cm

**1. Area of the circular bases:**
The formula to calculate the area of a circle is given by:
A = πr², where r is the radius of the circle.

The radius of the circular base is 10.5 cm, so the area of one circular base is:
A₁ = π(10.5)²

Since there are two circular bases, the total area of the circular bases is:
A_b = 2A₁

**2. Curved Surface Area:**
The curved surface area of a cylinder is given by the formula:
CSA = 2πrh, where r is the radius of the base and h is the height of the cylinder.

In this case, the height of the cylinder is given as the length, which is 18 cm.

The curved surface area is:
CSA = 2π(10.5)(18)

**3. Total Surface Area:**
The total surface area is the sum of the area of the circular bases and the curved surface area.

Total Surface Area = A_b + CSA

Substituting the values calculated above:
Total Surface Area = 2A₁ + CSA

**Calculations:**

1. Area of the circular bases:
A₁ = π(10.5)²

2. Curved Surface Area:
CSA = 2π(10.5)(18)

3. Total Surface Area:
Total Surface Area = 2A₁ + CSA

Now let's calculate the values:

A₁ = π(10.5)²
A₁ = 346.5π

CSA = 2π(10.5)(18)
CSA = 378π

Total Surface Area = 2A₁ + CSA
Total Surface Area = 2(346.5π) + 378π
Total Surface Area = 693π + 378π
Total Surface Area = 1071π

Approximating the value of π to 3.14:
Total Surface Area ≈ 1071 * 3.14
Total Surface Area ≈ 3363.94 cm²

Rounding off to the nearest whole number, the total surface area is approximately 3364 cm².

Therefore, the correct option is C) 1881 cm².

To find the width of the beam, we need to calculate the cross-sectional area of the wood beam and then divide the volume of wood by this area.

1. Calculate the cross-sectional area of the wood beam:
Given:
Length of the beam = 5 m
Width of the beam = ?
Height of the beam = 36 cm = 0.36 m

The cross-sectional area of the beam can be calculated using the formula:
Area = Length × Width

Substituting the given values:
Area = 5 m × Width

2. Convert the volume of wood to cubic meters:
Given:
Volume of wood = 1.35 m^3

3. Divide the volume of wood by the cross-sectional area to find the width:
Width = Volume of wood / Area = 1.35 m^3 / (5 m × Width)

To simplify the calculation, we can convert the volume of wood and the width to cubic centimeters (cm^3) and centimeters (cm), respectively. This can be done by multiplying by 100^3 since 1 m = 100 cm.

Converting the volume of wood to cm^3:
Volume of wood = 1.35 m^3 × (100 cm/m)^3 = 1.35 × 10^6 cm^3

Converting the width to cm:
Width = ? cm

Now, substituting the values into the equation:
Width = (1.35 × 10^6 cm^3) / (5 m × Width)

Simplifying the equation:
Width = 270,000 cm^2 / Width

To solve for the width, we can multiply both sides of the equation by Width:
Width^2 = 270,000 cm^2

Taking the square root of both sides:
Width = √(270,000 cm^2)

Width ≈ 519.62 cm

Therefore, the width of the beam is approximately 519.62 cm, which is not one of the given options (a, b, c, or d). Hence, the correct answer is "None of these."

Understanding the Problem
To find the volume of a cube given its total surface area, we need to utilize the formulas related to cubes.

Formulas to Remember
- The total surface area (TSA) of a cube is calculated using the formula:
TSA = 6a²
where 'a' is the length of one side of the cube.
- The volume (V) of a cube is calculated using the formula:
V = a³

Calculating the Side Length
Given that the total surface area is 486 cm², we can set up the equation:
6a² = 486
Now, we can solve for 'a':
a² = 486 / 6
a² = 81
a = √81
a = 9 cm

Calculating the Volume
Now that we have the side length 'a', we can find the volume:
V = a³
V = 9³
V = 729 cm³

Conclusion
The volume of the cube is 729 cm³, which corresponds to option 'A'.

Understanding the Problem
To determine how many planks can be cut from a wooden block, we need to compare the volumes of the block and the planks.
Dimensions of the Wooden Block
- Length: 5 m = 500 cm
- Breadth: 70 cm
- Thickness: 32 cm
Volume of the Wooden Block
- Volume = Length × Breadth × Thickness
- Volume = 500 cm × 70 cm × 32 cm = 1,120,000 cm³
Dimensions of One Plank
- Length: 2 m = 200 cm
- Breadth: 25 cm
- Thickness: 8 cm
Volume of One Plank
- Volume = Length × Breadth × Thickness
- Volume = 200 cm × 25 cm × 8 cm = 40,000 cm³
Calculating the Number of Planks
To find out how many planks can be prepared from the block:
- Number of Planks = Volume of Wooden Block / Volume of One Plank
- Number of Planks = 1,120,000 cm³ / 40,000 cm³ = 28
Conclusion
Thus, the total number of planks that can be prepared from the wooden block is 28. Therefore, the correct answer is option 'A'.

To find the depth of the cylindrical tank, we can use the formula for the volume of a cylinder.

Volume of a Cylinder
The volume \( V \) of a cylinder is calculated using the formula:
\[ V = \pi r^2 h \]
Where:
- \( V \) = volume of the cylinder
- \( r \) = radius of the cylinder
- \( h \) = height (or depth) of the cylinder

Given Data
- Capacity \( V = 5632 \, m^3 \)
- Diameter \( d = 16 \, m \)

Calculating Radius
To find the radius, we use the diameter:
\[ r = \frac{d}{2} = \frac{16}{2} = 8 \, m \]

Substituting Values
Now, we substitute the values into the volume formula:
\[ 5632 = \pi (8)^2 h \]
Calculating \( (8)^2 \):
\[ (8)^2 = 64 \]
So the equation becomes:
\[ 5632 = \pi \times 64 \times h \]

Solving for Depth (h)
Now, we isolate \( h \):
\[ h = \frac{5632}{\pi \times 64} \]
Using \( \pi \approx 3.14 \):
\[ h = \frac{5632}{3.14 \times 64} \]
Calculating \( 3.14 \times 64 \):
\[ 3.14 \times 64 \approx 200.96 \]
Now substituting back:
\[ h \approx \frac{5632}{200.96} \approx 28 \, m \]

Conclusion
Thus, the depth of the cylindrical tank is approximately 28 meters. Therefore, the correct answer is option 'C'.

Let the ratio of the volumes of two cylinder be v1 and v2.

Understanding the Problem
To determine how much earth must be dug out to sink a well, we need to calculate the volume of a cylindrical shape created by the well.
Formula for Volume of a Cylinder
The formula for the volume (V) of a cylinder is:
V = π × r² × h
Where:
- r = radius of the base of the cylinder
- h = height (depth) of the cylinder
- π (Pi) is approximately 3.14
Given Data
- Depth of the well (h) = 22.5 m
- Diameter of the well = 7 m
Calculating the Radius
- Radius (r) = Diameter / 2 = 7 m / 2 = 3.5 m
Calculating the Volume
Now, substituting the values into the volume formula:
- V = π × (3.5 m)² × 22.5 m
Calculating step-by-step:
1. Calculate the radius squared:
- (3.5 m)² = 12.25 m²
2. Multiply by the height:
- 12.25 m² × 22.5 m = 275.625 m³
3. Now, multiply by π (approximately 3.14):
- V ≈ 3.14 × 275.625 m³ = 866.25 m³
Final Conclusion
The total volume of earth that must be dug out to sink the well is approximately 866.25 cubic meters.
Correct Answer
Therefore, the correct answer is option 'A' - 866.25 cu m.