Formula for the Volume of a Cylinder

Oct 13, 2025 by magicalceylontours.com 130 views

The Radius and Height of the Cylinder are in the Ratio 5:2: Finding the TSA of the Cylinder

In this article, we will explore the concept of finding the total surface area (TSA) of a cylinder when the ratio of its radius and height is given. We will use the formula for the volume of a cylinder to find the values of the radius and height, and then use these values to calculate the TSA.

The problem states that the radius and height of the cylinder are in the ratio 5:2. This means that if the radius is 5x, the height will be 2x. The volume of the cylinder is given as 8316cm³. We need to find the TSA of the cylinder.

Formula for the Volume of a Cylinder

The formula for the volume of a cylinder is:

V = πr²h

where V is the volume, π is a constant (approximately 3.14), r is the radius, and h is the height.

Using the Ratio to Find the Values of Radius and Height

Since the radius and height are in the ratio 5:2, we can write:

r = 5x h = 2x

Substituting these values into the formula for the volume of a cylinder, we get:

8316 = π(5x)²(2x)

Simplifying the equation, we get:

8316 = 50πx³

Now, we can solve for x:

x³ = 8316 / (50π) x³ = 53.04 x = 3.98 (approximately)

Finding the Values of Radius and Height

Now that we have the value of x, we can find the values of the radius and height:

r = 5x r = 5(3.98) r = 19.9 (approximately)

h = 2x h = 2(3.98) h = 7.96 (approximately)

Calculating the TSA of the Cylinder

The formula for the TSA of a cylinder is:

TSA = 2πr(h + r)

Substituting the values of r and h, we get:

TSA = 2π(19.9)(7.96 + 19.9) TSA = 2π(19.9)(27.86) TSA = 2194.5 (approximately)

In this article, we used the formula for the volume of a cylinder to find the values of the radius and height, and then used these values to calculate the TSA. We found that the TSA of the cylinder is approximately 2194.5 cm².

Here is the step-by-step solution to the problem:

Write the formula for the volume of a cylinder: V = πr²h
Substitute the values of r and h into the formula: 8316 = π(5x)²(2x)
Simplify the equation: 8316 = 50πx³
Solve for x: x³ = 8316 / (50π)
Find the value of x: x = 3.98 (approximately)
Find the values of r and h: r = 5x, h = 2x
Calculate the TSA of the cylinder: TSA = 2πr(h + r)

When solving this problem, there are a few common mistakes to avoid:

Not using the correct formula for the volume of a cylinder
Not substituting the values of r and h into the formula correctly
Not simplifying the equation correctly
Not solving for x correctly
Not finding the values of r and h correctly
Not calculating the TSA correctly

Here are a few practice problems to help you practice solving this type of problem:

The radius and height of a cylinder are in the ratio 3:4. If the volume of the cylinder is 1000cm³, find the TSA of the cylinder.
The radius and height of a cylinder are in the ratio 2:5. If the volume of the cylinder is 2000cm³, find the TSA of the cylinder.
The radius and height of a cylinder are in the ratio 4:3. If the volume of the cylinder is 3000cm³, find the TSA of the cylinder.

This problem has several real-world applications, including:

Engineering: When designing a cylinder, such as a pipe or a tank, it is essential to calculate the TSA to ensure that it is strong enough to withstand the forces acting on it.
Architecture: When designing a building, such as a skyscraper or a stadium, it is essential to calculate the TSA of the cylinder to ensure that it is strong enough to withstand the forces acting on it.
Physics: When studying the motion of a cylinder, such as a rolling ball or a spinning top, it is essential to calculate the TSA to understand the forces acting on it.

In conclusion, finding the TSA of a cylinder when the ratio of its radius and height is given is a complex problem that requires careful calculation and attention to detail. By following the steps outlined in this article, you can solve this problem and apply it to real-world situations.
Q&A: Finding the TSA of a Cylinder when the Ratio of its Radius and Height is Given

In our previous article, we explored the concept of finding the total surface area (TSA) of a cylinder when the ratio of its radius and height is given. We used the formula for the volume of a cylinder to find the values of the radius and height, and then used these values to calculate the TSA. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q: What is the formula for the TSA of a cylinder?

A: The formula for the TSA of a cylinder is:

TSA = 2πr(h + r)

where TSA is the total surface area, π is a constant (approximately 3.14), r is the radius, and h is the height.

Q: How do I find the values of the radius and height when the ratio is given?

A: To find the values of the radius and height when the ratio is given, you can use the following steps:

Write the formula for the volume of a cylinder: V = πr²h
Substitute the values of r and h into the formula: V = π(5x)²(2x)
Simplify the equation: V = 50πx³
Solve for x: x³ = V / (50π)
Find the value of x: x = ∛(V / (50π))
Find the values of r and h: r = 5x, h = 2x

Q: What is the significance of the ratio of the radius and height?

A: The ratio of the radius and height is significant because it affects the shape and size of the cylinder. A higher ratio of radius to height means that the cylinder will be more rounded and less tall, while a lower ratio means that the cylinder will be more elongated and less rounded.

Q: Can I use this method to find the TSA of a cylinder with a different shape?

A: No, this method is specifically designed for finding the TSA of a cylinder with a circular cross-section. If you have a cylinder with a different shape, such as a rectangular or triangular cross-section, you will need to use a different method to find the TSA.

Q: What are some common mistakes to avoid when finding the TSA of a cylinder?

A: Some common mistakes to avoid when finding the TSA of a cylinder include:

Not using the correct formula for the TSA of a cylinder
Not substituting the values of r and h into the formula correctly
Not simplifying the equation correctly
Not solving for x correctly
Not finding the values of r and h correctly
Not calculating the TSA correctly

Q: Can I use this method to find the TSA of a cylinder with a negative radius or height?

A: No, this method is designed for finding the TSA of a cylinder with a positive radius and height. If you have a cylinder with a negative radius or height, you will need to use a different method to find the TSA.

Q: What are some real-world applications of finding the TSA of a cylinder?

A: Some real-world applications of finding the TSA of a cylinder include:

Engineering: When designing a cylinder, such as a pipe or a tank, it is essential to calculate the TSA to ensure that it is strong enough to withstand the forces acting on it.
Architecture: When designing a building, such as a skyscraper or a stadium, it is essential to calculate the TSA of the cylinder to ensure that it is strong enough to withstand the forces acting on it.
Physics: When studying the motion of a cylinder, such as a rolling ball or a spinning top, it is essential to calculate the TSA to understand the forces acting on it.

In conclusion, finding the TSA of a cylinder when the ratio of its radius and height is given is a complex problem that requires careful calculation and attention to detail. By following the steps outlined in this article and avoiding common mistakes, you can solve this problem and apply it to real-world situations.