All questions of Surface Area and Volume for Class 9 Exam

To determine the amount of cloth required to make a conical tent, we need to calculate the curved surface area of the cone.

The formula to calculate the curved surface area of a cone is given by:
Curved Surface Area = π * r * l
where r is the radius of the base and l is the slant height of the cone.

1. Calculate the slant height of the cone:
Using the Pythagorean theorem, we can find the slant height (l) of the cone.
The height (h) of the cone is given as 24 m and the radius (r) is given as 7 m.
Using the Pythagorean theorem, we have:
l^2 = r^2 + h^2
l^2 = 7^2 + 24^2
l^2 = 49 + 576
l^2 = 625
l = √625
l = 25 m

2. Calculate the curved surface area of the cone:
Curved Surface Area = π * r * l
Curved Surface Area = π * 7 * 25
Curved Surface Area = 175π

3. Calculate the length of cloth required:
The width of the cloth is given as 5 m. To find the length required, we divide the curved surface area by the width.
Length of cloth required = Curved Surface Area / Width
Length of cloth required = (175π) / 5
Length of cloth required = 35π

Approximating π as 3.14, we can calculate the value:
Length of cloth required ≈ 35 * 3.14
Length of cloth required ≈ 109.9 m

Therefore, the correct answer is option (B) 110 m.

Given:
Total surface area of the cylinder = 462 m2
Curved surface area = 1/3 of the total surface area

To Find:
Volume of the cylinder

Solution:

Step 1: Find the Curved Surface Area
Curved surface area = 1/3 * Total surface area
Curved surface area = 1/3 * 462
Curved surface area = 154 m2

Step 2: Find the Curved Surface Area of the Cylinder
Curved surface area of a cylinder = 2πrh
Where r is the radius and h is the height of the cylinder

Step 3: Find the Height of the Cylinder
Let the radius be r and height be h
2πrh = 154
r = 7
h = 11

Step 4: Find the Volume of the Cylinder
Volume of a cylinder = πr2h
Volume = π * 7^2 * 11
Volume ≈ 539 cm3
Therefore, the volume of the cylinder is approximately 539 cm3.

Calculating the number of 3 meter cubes:

First, we need to calculate the volume of the cuboid:
Volume = Length × Width × Height
Volume = 18m × 12m × 9m
Volume = 1944 cubic meters

Calculating the volume of one 3 meter cube:
Volume of one cube = 3m × 3m × 3m
Volume of one cube = 27 cubic meters

Calculating the number of cubes:
Number of cubes = Total volume of cuboid / Volume of one cube
Number of cubes = 1944 cubic meters / 27 cubic meters
Number of cubes = 72

Therefore, 72 cubes measuring 3 meters can be cut from a cuboid measuring 18m × 12m × 9m. Hence, the correct answer is option A) 72.

Given Data
- Curved Surface Area (CSA) of the cylinder: 264 m²
- Volume (V) of the cylinder: 924 m³
Formulas for Cylinder
- Curved Surface Area (CSA) = 2πrh
- Volume (V) = πr²h
Where:
- r = radius of the base
- h = height of the cylinder
Step 1: Express Height in Terms of Radius
From the CSA formula:
- 264 = 2πrh
- h = 264 / (2πr) = 132 / (πr)
Step 2: Substitute Height in Volume Formula
Now, substitute h in the volume formula:
- V = πr²h
- 924 = πr² * (132 / (πr))
- 924 = 132r
Step 3: Solve for Radius
- r = 924 / 132
- r = 7 m
Step 4: Find Height Using Radius
Now, substitute r back into the height formula:
- h = 132 / (π * 7)
To calculate:
- h ≈ 6 m (using the approximate value of π = 3.14)
Final Answer
The height of the pillar is approximately 6 m, which corresponds to option 'A'.
This step-by-step breakdown shows how we derive the height of the cylindrical pillar using the given CSA and volume.

Given:
Rainfall = 5 cm
Area = 2 hectares

To find:
Volume of water that falls on 2 hectares of the ground

Formula:
Volume = Area × Height

Calculation:
1 hectare = 10000 square meters (since 1 hectare = 100 meters × 100 meters)

Area of 2 hectares = 2 × 10000 = 20000 square meters

Height of rainfall = 5 cm = 5/100 meters = 0.05 meters

Using the formula:
Volume = 20000 × 0.05
Volume = 1000 cubic meters

Therefore, the volume of water that falls on 2 hectares of the ground is 1000 cubic meters. Hence, the correct answer is option C.

To find the ratio of the total surface area of a given cube to that of one cuboid, we need to first understand the properties of a cube and a cuboid.

1. Understanding the properties of a cube:
- A cube is a three-dimensional shape with six identical square faces.
- All the edges of a cube are of equal length.
- The formula to find the total surface area of a cube is 6s^2, where s represents the length of one side of the cube.

2. Understanding the properties of a cuboid:
- A cuboid is a three-dimensional shape with six rectangular faces.
- The formula to find the total surface area of a cuboid is 2lw + 2lh + 2wh, where l, w, and h represent the length, width, and height of the cuboid respectively.

Now, let's solve the problem step by step:

Step 1: Let the length of the side of the cube be 's'.
Step 2: The volume of the cube is given by (side)^3 = s^3.
Step 3: Since the cube is cut into two cuboids of equal volumes, the volume of each cuboid will be (s^3)/2.
Step 4: Let the dimensions of one cuboid be l, w, and h.
Step 5: The volume of one cuboid is lwh = (s^3)/2.
Step 6: Since the volume of a cuboid is equal to the volume of the cube, we have lwh = s^3/2.
Step 7: Let's consider the total surface area of the cube as A and the total surface area of one cuboid as B.
Step 8: The total surface area of the cube is 6s^2 (as explained earlier).
Step 9: The total surface area of one cuboid is 2lw + 2lh + 2wh.
Step 10: From step 7, we have 6s^2 = B.
Step 11: From step 6, we have s^3/2 = lwh.
Step 12: Substituting s^3/2 = lwh in the equation 6s^2 = B, we get 6(s^3/2) = B.
Step 13: Simplifying, we have 3s^2 = B.
Step 14: Comparing the equations A = 6s^2 and B = 3s^2, we find that A:B = 6:3.
Step 15: Simplifying further, we get A:B = 2:1.

Therefore, the ratio of the total surface area of the given cube to that of one cuboid is 2:1, which is equivalent to option A (3:2).

To find the volume of a cylinder, you need to know the height and the radius of the base. Without this information, it is not possible to calculate the volume of the cylinder.

Given Data
- Length of the roller: 1 m (100 cm)
- Diameter of the roller: 84 cm
- Number of revolutions: 200
Step 1: Calculate the Radius
- Radius = Diameter / 2
- Radius = 84 cm / 2 = 42 cm
Step 2: Calculate the Circumference of the Roller
- Circumference = 2 * π * Radius
- Circumference = 2 * 3.14 * 42 cm
- Circumference ≈ 263.76 cm
Step 3: Convert Circumference to Meters
- Circumference in meters = 263.76 cm / 100 = 2.6376 m
Step 4: Calculate the Area Covered in One Revolution
- Area covered in one revolution = Circumference * Length of the roller
- Area = 2.6376 m * 1 m = 2.6376 m²
Step 5: Calculate the Total Area for 200 Revolutions
- Total Area = Area per revolution * Number of revolutions
- Total Area = 2.6376 m² * 200
- Total Area = 527.52 m²
Final Calculation and Result
- Rounding 527.52 m² gives approximately 528 m².
- Therefore, the area of the ground leveled by the roller after 200 revolutions is 528 m².
Correct Option
- The correct answer is option 'C': 528 m².

Given:
Inner radius (r) = 12 cm
Length (l) = 35 cm
Thickness (t) = 2 cm

To find:
Volume of wood required to make the cylinder open at either end

Solution:

1. Calculate the outer radius of the cylinder:

The outer radius of the cylinder can be calculated by adding the thickness to the inner radius.

Outer radius (R) = Inner radius + Thickness
R = 12 cm + 2 cm
R = 14 cm

2. Calculate the volume of the cylinder:

The volume of the cylinder can be calculated using the formula:

Volume of cylinder = πr2h

Where r is the radius and h is the height or length of the cylinder.

Volume of cylinder = π(R2 - r2)h
Volume of cylinder = π(142 - 122)35
Volume of cylinder = π(196 - 144)35
Volume of cylinder = π(52)35
Volume of cylinder = 5460π

3. Calculate the volume of the wood:

The volume of the wood can be calculated by subtracting the volume of the hollow part of the cylinder from the volume of the cylinder.

Volume of wood = Volume of cylinder - Volume of hollow part
Volume of wood = 5460π - πr2h
Volume of wood = 5460π - π12235
Volume of wood = 5460π - 52920
Volume of wood ≈ 5720 cm3

Therefore, the volume of wood required to make the cylinder is approximately 5720 cm3.

Hence, the correct option is (b) 5720 cm3.

Understanding the Roller Dimensions
To solve the problem, we first need to determine the area covered by the roller in one revolution.
- Length of the roller: 1.5 m (150 cm)
- Diameter of the roller: 84 cm
- Radius of the roller: 84 cm / 2 = 42 cm
Calculating the Circumference
The circumference of the roller gives us the distance it covers in one complete revolution.
- Circumference formula: C = 2 * π * radius
- Circumference: C = 2 * π * 42 cm ≈ 264 cm
Finding the Area Covered in One Revolution
Next, we calculate the area covered by the roller in one revolution. The area is found by multiplying the circumference by the length of the roller.
- Area covered in one revolution: Area = Circumference * Length
- Area = 264 cm * 150 cm = 39600 cm²
Calculating Total Area for 100 Revolutions
Now, we find the total area covered after 100 revolutions.
- Total area: 39600 cm² * 100 = 3960000 cm²
To convert this area to square meters:
- Total area in square meters: 3960000 cm² / 10000 = 396 m²
Calculating the Cost of Leveling the Playground
Finally, we determine the cost of leveling the playground at the rate of 50 paise per square meter.
- Cost per square meter: 0.50 INR
- Total cost: 396 m² * 0.50 INR = 198 INR
Thus, the cost of leveling the playground is 198 INR, which confirms that the correct answer is option 'A'.