Surface Areas and Volumes Class 10th Important Questions with Answers Mathematics

Given:
Length = 5 meters
Breadth = 4 meters
Height = 3 meters
Cost of whitewashing = Rs. 7.50 per square meter

Step 1:-Find the area of the four walls.

Formula: Area = 2 × height × (length + breadth)
Area = 2 × 3 × (5 + 4)
Area = 6 × 9 = 54 square meters

Step 2:-Calculate the cost of whitewashing.

Cost = Area × Rate
Cost = 54 × 7.50 = Rs. 405.00

Hence cost is Rs. 405.00

Given:
Perimeter of floor = 250 meters
Cost of whitewashing = Rs. 15,000
Rate = Rs. 10 per square meter

Step 1: Calculate the area of the four walls

Let the height of the room be h meters.

Area of four walls = Perimeter × Height
=>  Area = 250 × h

Step 2: Use the cost to find the area

Cost = Area × Rate
15000 = (250 × h) × 10

Step 3: Solve for h

15000 = 2500 × h
h = 15000 / 2500
h = 6

Hence height of the room is6 meters

Given:
Edge of each cube = 12 cm
When two cubes are joined end to end, the new shape is a cuboid.

Step 1: Find the dimensions of the cuboid

Length = 12 cm + 12 cm = 24 cm
Breadth = 12 cm
Height = 12 cm

Step 2: Use the surface area formula for a cuboid

Surface Area = 2 × (length × breadth + breadth × height + height × length)
Surface Area = 2 × (24 × 12 + 12 × 12 + 12 × 24)
Surface Area = 2 × (288 + 144 + 288)
Surface Area = 2 × 720 = 1440 cm²

Hence surface area of the new cuboid  is 1440 cm²

Length of the pipe = 77 cm
Inner diameter = 4 cm 
Then Inner radius of pipe is = 4 ÷ 2 = 2 cm

Formula for curved surface area (CSA) of a cylinder:

CSA = 2 × π × r × h
Where:
r = radius = 2 cm
h = height (length of pipe) = 77 cm
π = 22/7

Substitute values then CSA is

CSA = 2 × (22/7) × 2 × 77
CSA = (4 × 22 × 77) / 7
CSA = (6776) / 7 = 968 cm²

Hence inner curved surface area is 968 cm²

Diameter of the roller = 84 cm
Then radius is = 84 ÷ 2 = 42 cm
Length of the roller = 120 cm
Number of revolutions = 500

Step 1:Calculate the curved surface area of the roller

The formula for the curved surface area (CSA) of a cylinder is:
CSA = 2 × π × r × h
Where:
r = 42 cm (radius)
h = 120 cm (length)

Substitute the values:
CSA = 2 × (22/7) × 42 × 120
CSA = 2 × (22/7) × 5040
CSA = (2 × 22 × 5040) ÷ 7 = 31680 cm²

Step 2:Calculate the area of the playground

Each revolution of the roller covers an area equal to its curved surface area.
So, the area covered by 500 revolutions is:
Total area = 500 × 31680 cm² = 15840000 cm²

Convert the area from cm² to m² (since 1 m² = 10,000 cm²):
Total area = 15840000 ÷ 10000 = 1584 m²

Hence the area of the playground is 1584 m².

Diameter of the pillar = 50 cm = 0.5 m (since 1 m = 100 cm)
Then radius = Diameter ÷ 2 = 0.5 ÷ 2 = 0.25 m
Height of the pillar = 3.5 m
Rate of painting = Rs. 12.50 per m²

Step 1: Calculate the curved surface area (CSA) of the cylinder

The formula for the curved surface area of a cylinder is:
CSA = 2πrh
Where:
r = radius = 0.25 m
h = height = 3.5 m

Substitute the values:
CSA = 2 × (22/7) × 0.25 × 3.5
CSA = 2 × (22/7) × 0.875
CSA = (44/7) × 0.875
CSA = 5.5 m²

Step 2: Calculate the cost of painting

Cost = CSA × Rate
Cost = 5.5 × 12.50
Cost = Rs. 68.75

Hence the cost of painting the curved surface is Rs. 68.75

Depth of the river = 3 meters
Width of the river = 40 meters
Speed of the river = 2 km per hour = 2000 meters per hour (since 1 km = 1000 meters)

Step 1:-Calculate the volume of water flowing per hour.

The volume of water flowing per hour can be calculated using the formula:
Volume = Length × Width × Depth

Here, the length is the distance the water flows in one hour, which is 2000 meters.
So, the volume of water flowing in 1 hour is:

Volume per hour = 2000 × 40 × 3 = 240,000 cubic meters

Step 2:- Convert the volume from per hour to per minute.

Since there are 60 minutes in an hour, the volume per minute is:
Volume per minute = 240,000 ÷ 60 = 4,000 cubic meters

Hence the volume of water flowing into the sea in one minute is 4,000 cubic meters

We are given:

Volume of the cylinder (V) = 6160 cm³
Diameter of the base = 28 m = 2800 cm  (1 m = 100 cm)
Then radius (r) = diameter ÷ 2 = 2800 ÷ 2 = 1400 cm

Step 1: - Use the formula for the volume of a cylinder
Volume (V) = π × r² × h
Where:
V = volume
r = radius
h = height (depth)

Substitute the known values:
6160 = π × (1400)² × h
6160 = π × 1,960,000 × h

Solve for h:
h = 6160 ÷ (π × 1,960,000)
=>  h = 6160 ÷ (3.1416 × 1,960,000)
=>  h = 6160 ÷ 6,157,536
=>  h = 0.001 cm

Hence depth of the tank is 0.001 cm

Given:
A cube of edge 14 cm.
A cone of maximum size is carved out from the cube.

Step 1: Dimensions of the Cone
Height of the cone (h) = 14 cm (same as the cube’s height)
Diameter of the cone’s base = 14 cm ⇒ Radius (r) = 7 cm

Step 2: Surface Area of the Cone
Total surface area (TSA) of a cone = πr(l + r)
Where: r = 7 cm  &  h = 14 cm
Then Slant height (l) = √(r² + h²) = √(49 + 196) = √245 = 7√5 = 15.652

Now,
Total surface area (TSA) of a cone = π × 7 × (7 + 7√5) = 49π(1 + √5)
Using π = 3.1416 and √5 =  2.236:

TSA   =  49 × 3.1416 × (1 + 2.236)
TSA   =  49 × 3.1416 × 3.236
TSA   =  498.1 cm²

Step 3: Surface Area of the Remaining Solid
Surface area of the cube = 6 × (14)² = 6 × 196 = 1176 cm²
Base area of cone = πr² = π × 49 = 153.94 cm²
Curved surface area of cone = πrl = π × 7 × 7√5 = 344.2 cm²

Now,
Remaining surface area = Surface area of cube - base area of cone + curved surface area of cone
Remaining surface area = 1176 - 153.94 + 344.2
Remaining surface area = 1366.26 cm²

Hence:
Surface area of the cone = 498.1 cm²
Surface area of the remaining solid = 1366.26 cm²

First, find the volume of the original metallic sphere.
The formula for the volume of a sphere is:
V = (4/3) × π × r³
Radius of the sphere = 10.5 cm

So, V = (4/3) × π × (10.5)³ = (4/3) × π × 1157.625  =  4851.375 cm³

Next, find the volume of one small cone.
The formula for the volume of a cone is:
V = (1/3) × π × r² × h
Radius = 3.5 cm, Height = 3 cm
V = (1/3) × π × (3.5)² × 3 = π × 12.25 ≈ 38.48 cm³

Now, divide the volume of the sphere by the volume of one cone:
Number of cones = 4851.375 ÷ 38.48 ≈ 126

Hence the number of cones formed 126 

Width of canal = 300 cm = 3 m
Depth of canal = 120 cm = 1.2 m
Speed of water = 20 km/h = 20,000 meters per hour = 333.33 meters per minute
Time = 20 minutes

Step 1: Find the volume of water flowing in 20 minutes
Cross-sectional area of canal = 3 m × 1.2 m = 3.6 m²
Distance water flows in 20 minutes = 333.33 × 20 = 6666.67 meters

Volume = area × distance = 3.6 × 6666.67 = 24,000 m³

Step 2: Find the irrigated area
Required water depth = 8 cm = 0.08 m
Area = volume ÷ depth = 24,000 ÷ 0.08 = 300,000 m²

Hence 300,000 m² will be irrigated.

Let the edges of the three cubes be in the ratio 3:4:5.
So, let the edges be 3x, 4x, and 5x.

Volume of a cube = edge³
Total volume of the three cubes = (3x)³ + (4x)³ + (5x)³
= 27x³ + 64x³ + 125x³ = 216x³

The three cubes are melted and formed into one new cube.
The diagonal of the new cube is given as 12√3 cm.

The formula for the diagonal of a cube = √3 × edge
So,
12√3 = √3 × edge
⇒ edge = 12 cm

Volume of the new cube = 12³ = 1728 cm³

Now, equate the volumes:
216x³ = 1728
⇒ x³ = 8
⇒ x = 2

Now, calculate the edges of the original cubes:
First cube: 3x = 3 × 2 = 6 cm
Second cube: 4x = 4 × 2 = 8 cm
Third cube: 5x = 5 × 2 = 10 cm

Hence the edges of the three cubes are 6 cm, 8 cm, and 10 cm.