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Geometry Problems for 10th Standard - Step by Step Solutions

Aluminium Sphere to Cylinder

An aluminium sphere of radius 12 cm is melted to make a cylinder of radius 8 cm. Find the height of the cylinder.

Step 1: Find Volume of Sphere

First, we calculate the volume of the original aluminium sphere using the sphere volume formula:

V_sphere = (4/3)πr³

Given radius (r) = 12 cm

V = (4/3) × π × (12)³ = (4/3) × π × 1728 = 2304π cm³

Step 2: Volume Remains Same

When the sphere is melted and recast into a cylinder, the volume remains the same.

V_sphere = V_cylinder = 2304π cm³

Step 3: Find Height of Cylinder

Now, use the cylinder volume formula to find the height (h):

V_cylinder = πr²h
2304π = π × (8)² × h
2304π = 64π × h
h = 2304π / 64π = 36 cm

Final Answer: The height of the cylinder is 36 cm

Water Flow into Rectangular Tank

Water is flowing at the rate of 15 km per hour through a pipe of diameter 14 cm into a rectangular tank which is 50 m long and 44 m wide. Find the time in which the level of water in the tanks will rise by 21 cm.

Step 1: Convert Units

First, convert all measurements to consistent units (meters):

Pipe diameter = 14 cm = 0.14 m → Radius (r) = 0.07 m
Flow rate = 15 km/h = 15,000 m/h

Step 2: Calculate Pipe Cross-Section

Find cross-sectional area of the pipe:

A_pipe = πr² = π × (0.07)² ≈ 0.0154 m²

Step 3: Calculate Flow Volume Rate

Volume flowing per hour (discharge rate):

Q = Area × Velocity = 0.0154 m² × 15,000 m/h = 231 m³/h

Step 4: Calculate Tank Volume Increase

Volume needed to raise level by 21 cm (0.21 m):

V = Length × Width × Height = 50 m × 44 m × 0.21 m = 462 m³

Step 5: Calculate Time Required

Time = Total Volume / Flow Rate:

Time = 462 m³ ÷ 231 m³/h = 2 hours

Final Answer: The required time is 2 hours

Conical Flask to Cylindrical Flask

A conical flask is full of water. The flask has base radius r units and height h units, the water is poured into a cylindrical flask of base radius ar units. Find the height of water in the cylindrical flask.

Step 1: Volume of Conical Flask

Calculate volume of water in conical flask:

V_cone = (1/3)πr²h

Step 2: Volume in Cylinder

This same volume fills the cylinder to height H:

V_cylinder = π(ar)² × H = πa²r²H

Step 3: Set Volumes Equal

Since volumes are equal:

(1/3)πr²h = πa²r²H
(1/3)h = a²H
H = h/(3a²)

Final Answer: The height of water is h/(3a²) units

Cone to Hollow Sphere

A solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. If the external diameter of the sphere is 10 cm, find the internal diameter.

Step 1: Volume of Cone

Calculate volume of the cone (r = 7 cm, h = 8 cm):

V = (1/3)πr²h = (1/3)π×7²×8 ≈ 130.67π cm³

Step 2: External Sphere Volume

External radius (R) = 5 cm:

V_external = (4/3)πR³ = (4/3)π×125 ≈ 166.67π cm³

Step 3: Internal Sphere Volume

Let internal radius = r. Volume of hollow part:

V_hollow = V_external - V_cone
(4/3)πr³ = 166.67π - 130.67π = 36π
r³ = 27 → r = 3 cm

Step 4: Internal Diameter

Internal diameter = 2 × internal radius:

d = 2 × 3 cm = 6 cm

Final Answer: Internal diameter is 6 cm

Overhead Tank and Sump

Seenu's house has an overhead tank (radius 60 cm, height 105 cm) filled from a cuboid sump (2m × 1.5m × 1m). Find water left in sump after filling tank.

Step 1: Sump Volume

Calculate initial volume of water in sump:

V_sump = 2m × 1.5m × 1m = 3 m³ = 3,000,000 cm³

Step 2: Tank Volume

Calculate volume of overhead tank (r = 60 cm, h = 105 cm):

V_tank = πr²h = π×60²×105 ≈ 1,188,000 cm³

Step 3: Remaining Water

Subtract tank volume from sump volume:

Remaining = 3,000,000 - 1,188,000 = 1,812,000 cm³ = 1.812 m³

Final Answer: Water left in sump is 1.812 m³

Hemisphere to Cylinder

A hollow hemisphere (internal diameter 6 cm, external 10 cm) is melted into a solid cylinder (diameter 14 cm). Find the cylinder's height.

Step 1: Hemisphere Volume

Calculate volume of hollow hemisphere material:

V = (2/3)π(R³ - r³) = (2/3)π(5³ - 3³) = (2/3)π(125 - 27) ≈ 65.33π cm³

Step 2: Cylinder Volume

This volume forms the cylinder (r = 7 cm):

V = πr²h → 65.33π = π×49×h
h = 65.33/49 ≈ 1.33 cm

Final Answer: Cylinder height is ≈1.33 cm

Sphere to Hollow Cylinder

A solid sphere (radius 6 cm) is melted into a hollow cylinder (external radius 5 cm, height 32 cm). Find the cylinder's thickness.

Step 1: Sphere Volume

Calculate volume of the sphere (r = 6 cm):

V = (4/3)πr³ = (4/3)π×216 = 288π cm³

Step 2: Hollow Cylinder Volume

Let internal radius = r. Volume of cylinder material:

V = πh(R² - r²) = 32π(25 - r²) = 288π
25 - r² = 9 → r² = 16 → r = 4 cm

Step 3: Calculate Thickness

Thickness = External radius - Internal radius:

Thickness = 5 cm - 4 cm = 1 cm

Final Answer: Cylinder thickness is 1 cm

Juice Transfer Percentage

A hemispherical bowl is poured into a cylindrical vessel (radius 50% more than height) with same diameter. Find percentage of juice transferred.

Step 1: Hemisphere Volume

Let radius of both = r. Hemisphere volume:

V_hemi = (2/3)πr³

Step 2: Cylinder Dimensions

Cylinder radius = r (same diameter), height = h
Given radius = 1.5 × height → r = 1.5h → h = (2/3)r

Step 3: Cylinder Volume

Calculate cylinder volume:

V_cyl = πr²h = πr² × (2/3)r = (2/3)πr³

Step 4: Calculate Percentage

Both volumes are equal!

V_hemi = V_cyl = (2/3)πr³
Percentage transferred = 100%

Final Answer: 100% of juice can be transferred

Congratulations!