Volume Of Cylinder Questions And Answers

Let's delve into the world of cylinders and conquer the concept of their volume through a series of questions and answers. Understanding how to calculate the volume of a cylinder is a fundamental skill in mathematics and has practical applications in various fields, from engineering to everyday life.

Unveiling the Cylinder: A Journey into Shapes

Before diving into calculations, let's solidify our understanding of what a cylinder actually is. Imagine a stack of identical circular discs placed perfectly on top of each other. That, in essence, forms a cylinder. More formally, a cylinder is a three-dimensional geometric shape with two parallel circular bases connected by a curved surface. Key characteristics of a cylinder include:

• Circular Bases: Two identical circles that form the top and bottom of the cylinder.
• Radius (r): The distance from the center of the circular base to any point on its circumference.
• Height (h): The perpendicular distance between the two circular bases.
• Axis: The imaginary line connecting the centers of the two circular bases.

The Volume Equation: Cracking the Code

The volume of any three-dimensional object represents the amount of space it occupies. For a cylinder, calculating its volume boils down to a single, elegant equation:

Volume (V) = πr²h

Where:

• V represents the volume of the cylinder.
• π (pi) is a mathematical constant approximately equal to 3.14159.
• r represents the radius of the circular base.
• h represents the height of the cylinder.

This equation tells us that the volume of a cylinder is directly proportional to the area of its circular base (πr²) and its height (h).

Question & Answer Session: Putting Theory into Practice

Let's solidify your understanding with a series of questions and step-by-step solutions.

Question 1:

A cylinder has a radius of 5 cm and a height of 10 cm. What is its volume?

Answer:

1. Identify the given values:
   • Radius (r) = 5 cm
   • Height (h) = 10 cm
2. Apply the formula:
   • V = πr²h
   • V = π * (5 cm)² * 10 cm
   • V = π * 25 cm² * 10 cm
   • V = 250π cm³
3. Approximate the value:
   • V ≈ 250 * 3.14159 cm³
   • V ≈ 785.40 cm³

Therefore, the volume of the cylinder is approximately 785.40 cubic centimeters.

Question 2:

A cylindrical water tank has a diameter of 4 meters and a height of 3 meters. How much water can the tank hold in cubic meters?

Answer:

1. Identify the given values:
   • Diameter = 4 meters
   • Height (h) = 3 meters
2. Calculate the radius:
   • Radius (r) = Diameter / 2 = 4 meters / 2 = 2 meters
3. Apply the formula:
   • V = πr²h
   • V = π * (2 m)² * 3 m
   • V = π * 4 m² * 3 m
   • V = 12π m³
4. Approximate the value:
   • V ≈ 12 * 3.14159 m³
   • V ≈ 37.70 m³

Therefore, the water tank can hold approximately 37.70 cubic meters of water.

Question 3:

The volume of a cylinder is 150π cubic inches, and its height is 6 inches. What is the radius of the cylinder?

Answer:

1. Identify the given values:
   • Volume (V) = 150π cubic inches
   • Height (h) = 6 inches
2. Apply the formula and rearrange to solve for r:
   • V = πr²h
   • 150π = πr² * 6
   • Divide both sides by 6π:
   • 150π / 6π = r²
   • 25 = r²
3. Take the square root of both sides:
   • √25 = r
   • r = 5 inches

Therefore, the radius of the cylinder is 5 inches.

Question 4:

A cylindrical can of soup has a radius of 3.5 cm and a volume of 385 cm³. What is the height of the can?

Answer:

1. Identify the given values:
   • Radius (r) = 3.5 cm
   • Volume (V) = 385 cm³
2. Apply the formula and rearrange to solve for h:
   • V = πr²h
   • 385 = π * (3.5)² * h
   • 385 = π * 12.25 * h
   • Divide both sides by (π * 12.25):
   • h = 385 / (π * 12.25)
3. Approximate the value:
   • h ≈ 385 / (3.14159 * 12.25)
   • h ≈ 385 / 38.4845
   • h ≈ 10 cm

Therefore, the height of the soup can is approximately 10 cm.

Question 5:

A solid metal cylinder is melted down and recast into a new cylinder with twice the radius. If the original cylinder had a height of 12 cm, what is the height of the new cylinder, assuming the volume remains the same?

Answer:

1. Let's define variables:
   • Original cylinder: radius = r, height = 12 cm, volume = V₁
   • New cylinder: radius = 2r, height = h₂, volume = V₂
2. Write the volume equations:
   • V₁ = πr² * 12
   • V₂ = π(2r)² * h₂ = π * 4r² * h₂
3. Since the volume remains the same (V₁ = V₂):
   • πr² * 12 = π * 4r² * h₂
4. Divide both sides by πr²:
   • 12 = 4 * h₂
5. Solve for h₂:
   • h₂ = 12 / 4
   • h₂ = 3 cm

Therefore, the height of the new cylinder is 3 cm.

Question 6:

A hollow cylindrical pipe has an outer radius of 8 cm, an inner radius of 6 cm, and a length of 20 cm. What is the volume of the material used to make the pipe?

Answer:

1. Understand the concept: The volume of the material is the difference between the volume of the outer cylinder and the volume of the inner cylinder (the hollow space).
2. Identify the given values:
   • Outer radius (R) = 8 cm
   • Inner radius (r) = 6 cm
   • Length (h) = 20 cm
3. Calculate the volume of the outer cylinder (Vouter):
   • Vouter = πR²h = π * (8 cm)² * 20 cm = π * 64 cm² * 20 cm = 1280π cm³
4. Calculate the volume of the inner cylinder (Vinner):
   • Vinner = πr²h = π * (6 cm)² * 20 cm = π * 36 cm² * 20 cm = 720π cm³
5. Calculate the volume of the material:
   • Vmaterial = Vouter - Vinner = 1280π cm³ - 720π cm³ = 560π cm³
6. Approximate the value:
   • Vmaterial ≈ 560 * 3.14159 cm³
   • Vmaterial ≈ 1759.29 cm³

Therefore, the volume of the material used to make the pipe is approximately 1759.29 cubic centimeters.

Question 7:

A cylinder is inscribed in a cube with side length 10 cm. What is the volume of the cylinder? (Assume the cylinder's bases are tangent to the cube's faces).

Answer:

1. Visualize the problem: Imagine a cylinder perfectly fitting inside a cube. The diameter of the cylinder's base is equal to the side length of the cube, and the height of the cylinder is also equal to the side length of the cube.
2. Identify the given values:
   • Side length of the cube = 10 cm
3. Determine the cylinder's dimensions:
   • Radius of the cylinder (r) = side length / 2 = 10 cm / 2 = 5 cm
   • Height of the cylinder (h) = side length = 10 cm
4. Apply the formula:
   • V = πr²h
   • V = π * (5 cm)² * 10 cm
   • V = π * 25 cm² * 10 cm
   • V = 250π cm³
5. Approximate the value:
   • V ≈ 250 * 3.14159 cm³
   • V ≈ 785.40 cm³

Therefore, the volume of the cylinder is approximately 785.40 cubic centimeters.

Question 8:

The ratio of the radius to the height of a cylinder is 2:3. If the volume of the cylinder is 192π cubic units, find the radius and height of the cylinder.

Answer:

1. Express the radius and height in terms of a variable:
   • Let the radius (r) be 2x
   • Let the height (h) be 3x
2. Apply the volume formula:
   • V = πr²h
   • 192π = π(2x)²(3x)
   • 192π = π(4x²)(3x)
   • 192π = 12πx³
3. Solve for x:
   • Divide both sides by 12π:
   • 192π / 12π = x³
   • 16 = x³
   • Take the cube root of both sides:
   • ∛16 = x
   • x = ∛(8*2) = 2∛2
4. Find the radius and height:
   • Radius (r) = 2x = 2 * 2∛2 = 4∛2 units
   • Height (h) = 3x = 3 * 2∛2 = 6∛2 units

Therefore, the radius of the cylinder is 4∛2 units, and the height of the cylinder is 6∛2 units. Approximating ∛2 ≈ 1.26, we get r ≈ 5.04 units and h ≈ 7.56 units.

Question 9:

A cylinder with a radius of 4 cm is filled with water to a height of 7 cm. A metal cube with side length 3 cm is submerged in the water. How much does the water level rise in the cylinder?

Answer:

1. Calculate the volume of the cube:
   • Volume of cube = side³ = 3 cm * 3 cm * 3 cm = 27 cm³
2. Recognize that the volume of the water displaced by the cube is equal to the volume of the cube. This displaced water causes the water level to rise.
3. Let the rise in water level be 'Δh'. The volume of the water that rises in the cylinder is given by πr²Δh.
4. Set the volume of displaced water equal to the volume of the cube:
   • πr²Δh = 27 cm³
   • π(4 cm)²Δh = 27 cm³
   • π(16 cm²)Δh = 27 cm³
5. Solve for Δh:
   • Δh = 27 cm³ / (π * 16 cm²)
   • Δh ≈ 27 / (3.14159 * 16) cm
   • Δh ≈ 27 / 50.2654 cm
   • Δh ≈ 0.537 cm

Therefore, the water level rises by approximately 0.537 cm.

Question 10:

A rectangular sheet of metal measuring 22 cm by 12 cm is rolled along its length to form a cylinder. Find the volume of the cylinder.

Answer:

1. Visualize the process: When the sheet is rolled along its length, the length becomes the circumference of the base of the cylinder, and the width becomes the height of the cylinder.
2. Identify the given values:
   • Length of the sheet (circumference of the base) = 22 cm
   • Width of the sheet (height of the cylinder) = 12 cm
3. Calculate the radius of the base:
   • Circumference = 2πr
   • 22 cm = 2πr
   • r = 22 cm / (2π)
   • r = 11 cm / π
4. Apply the volume formula:
   • V = πr²h
   • V = π * (11/π cm)² * 12 cm
   • V = π * (121/π² cm²) * 12 cm
   • V = (121 * 12) / π cm³
   • V = 1452 / π cm³
5. Approximate the value:
   • V ≈ 1452 / 3.14159 cm³
   • V ≈ 462.11 cm³

Therefore, the volume of the cylinder is approximately 462.11 cubic centimeters.

Diving Deeper: Real-World Applications

The ability to calculate the volume of a cylinder isn't just an abstract mathematical exercise. It has numerous practical applications:

• Engineering: Engineers use this calculation to determine the capacity of cylindrical tanks, pipes, and other structures. This is crucial in designing efficient and safe systems for storing and transporting liquids and gases.
• Manufacturing: Manufacturers need to calculate the volume of materials required to produce cylindrical objects like cans, containers, and machine parts. This helps them optimize material usage and minimize waste.
• Construction: Calculating the volume of cylindrical columns or supports is essential for ensuring structural integrity in buildings and bridges.
• Everyday Life: From estimating the amount of paint needed to cover a cylindrical pillar to determining how much water a cylindrical glass can hold, the volume calculation finds its way into various everyday scenarios.
• Medicine: In medical imaging, understanding the volume of cylindrical or near-cylindrical structures (like blood vessels) is critical for diagnosis and treatment planning.

Common Pitfalls and How to Avoid Them

While the volume formula is straightforward, here are some common errors to watch out for:

• Using Diameter Instead of Radius: The formula requires the radius, not the diameter. Remember to divide the diameter by 2 before plugging it into the equation.
• Incorrect Units: Ensure that all measurements are in the same units before calculating the volume. If the radius is in centimeters and the height is in meters, convert one of them before applying the formula.
• Forgetting to Square the Radius: A common mistake is to multiply π by r and then by h, instead of multiplying by r². Remember the order of operations!
• Approximation Errors: While using 3.14 for π is often sufficient, for more accurate results, use a calculator with a π button or a more precise approximation like 3.14159.
• Confusing Volume with Surface Area: Volume measures the space an object occupies, while surface area measures the total area of its outer surfaces. Make sure you understand which one you need to calculate.

Mastering the Cylinder: Tips and Tricks

• Visualize the Problem: Before you start calculating, take a moment to visualize the cylinder and understand its dimensions. This can help you avoid mistakes.
• Draw a Diagram: If the problem isn't clear, draw a simple diagram of the cylinder and label its radius and height.
• Write Down the Formula: Writing down the formula (V = πr²h) before you start plugging in values can help you remember the correct steps.
• Check Your Units: Always double-check that your units are consistent before calculating.
• Estimate the Answer: Before you calculate, try to estimate the answer. This can help you catch any major errors.
• Practice, Practice, Practice: The best way to master the volume of a cylinder is to practice solving problems. Work through as many examples as you can find.
• Use Online Resources: There are many helpful online resources, such as calculators and tutorials, that can help you learn more about the volume of a cylinder.

The Enduring Relevance of Cylindrical Calculations

From the humble soup can on your shelf to the massive pipelines transporting resources across continents, cylinders are ubiquitous in our world. A solid understanding of their volume is therefore more than just an academic exercise; it's a key to unlocking a deeper understanding of the world around us. By mastering the formula and practicing with a variety of problems, you can confidently tackle any cylinder-related volume calculation that comes your way. Keep practicing, and you'll find that calculating cylinder volumes becomes second nature!