Cylinder Radius & Generator Calculation: A Step-by-Step Guide

Hey guys! Today, we're diving into a cool math problem involving cylinders. Specifically, we're going to figure out how to calculate the radius and generator (which is just a fancy word for height) of a right circular cylinder when we know some relationships between them and the diagonal of its axial section. Sounds intimidating? Nah, we'll break it down step by step so it's super easy to follow. Let's get started!

Understanding the Problem

Okay, so the problem states that we have a right circular cylinder. Remember what that is? It's basically a can – a shape with two circular ends and straight sides. Now, here's the key info:

• The radius (r) and the generator (g) are directly proportional to 4 and 6, respectively.
• The diagonal of the axial section is 30 cm.

What does all this mean? Let's unpack it.

Direct Proportionality

When we say that the radius and generator are directly proportional to 4 and 6, it means that their ratio is constant. In other words:

r / g = 4 / 6

We can simplify this to:

r / g = 2 / 3

This tells us that for every 2 units of radius, there are 3 units of generator (height). We can express this relationship using a constant of proportionality, k:

r = 2k g = 3k

So, both the radius and the generator are expressed in terms of this single variable k. This is super helpful because it reduces the number of unknowns we need to solve for.

Axial Section and its Diagonal

The axial section of a cylinder is the rectangle you get when you slice the cylinder straight down the middle, through its axis. The sides of this rectangle are the diameter of the base (2r) and the generator (g). The diagonal of this rectangle is what the problem tells us is 30 cm. This diagonal, the diameter and generator of our cylinder form a right triangle. The Pythagorean theorem is important here. Using the Pythagorean theorem we know that the square of the diagonal is equal to the sum of the squares of the other two sides.

Solving for the Radius and Generator

Now that we've understood the problem, let's solve it! Here's how we'll do it:

1. Use the Pythagorean Theorem: As we established earlier, the diagonal of the axial section, the diameter, and the generator form a right triangle. Therefore, we can use the Pythagorean theorem:

   (2r)^2 + g^2 = 30^2

2. Substitute the Proportionality Relationships: We know that r = 2k and g = 3k. Let's substitute these into the Pythagorean equation:

   (2 * 2k)^2 + (3k)^2 = 30^2 (4k)^2 + (3k)^2 = 900 16k^2 + 9k^2 = 900

3. Solve for k: Combine the terms and solve for k:

   25k^2 = 900 k^2 = 900 / 25 k^2 = 36 k = √36 k = 6

   Since k represents a length, we only take the positive square root.

4. Calculate the Radius and Generator: Now that we know k, we can find the radius and generator:

   r = 2k = 2 * 6 = 12 cm g = 3k = 3 * 6 = 18 cm

Therefore, the radius of the cylinder is 12 cm, and the generator (height) is 18 cm.

Verification

Just to be sure, let's plug these values back into the Pythagorean theorem to check our answer:

(2 * 12)^2 + 18^2 = 30^2 24^2 + 18^2 = 900 576 + 324 = 900 900 = 900

It checks out! Our calculations are correct.

Key Takeaways

• Direct Proportionality: Understanding direct proportionality is crucial for setting up the relationships between the radius and generator.
• Pythagorean Theorem: Recognizing the right triangle formed by the diagonal, diameter, and generator allows us to use the Pythagorean theorem to solve for the unknowns.
• Substitution: Substituting the proportionality relationships into the Pythagorean equation simplifies the problem and allows us to solve for a single variable.

Additional Tips and Considerations

• Units: Always remember to include the units (in this case, cm) in your final answer.
• Diagram: Drawing a diagram of the cylinder and its axial section can help you visualize the problem and identify the relevant relationships.
• Generalization: This approach can be applied to similar problems involving other geometric shapes and their properties.

Now that we've conquered this cylinder problem, you're well-equipped to tackle similar challenges! Keep practicing, and remember to break down complex problems into smaller, more manageable steps. And always double-check your work!

Real-World Applications

While this might seem like just a theoretical math problem, understanding cylinder calculations has real-world applications in various fields:

• Engineering: Engineers use these calculations to design and analyze cylindrical structures, such as pipes, tanks, and pressure vessels.
• Architecture: Architects need to understand cylinder geometry when designing buildings with cylindrical elements, such as columns or domes.
• Manufacturing: Manufacturing processes often involve cylindrical shapes, and accurate calculations are essential for ensuring proper dimensions and tolerances.
• Packaging: Designing cylindrical packaging requires knowing the surface area and volume of cylinders to optimize material usage and storage space.

Practice Problems

Want to put your newfound skills to the test? Here are a couple of practice problems:

1. In a right circular cylinder, the radius and generator are directly proportional to 5 and 8. The diagonal of the axial section is 39 cm. Find the radius and generator.
2. The radius and height of a cylinder are in the ratio 3:5. If the diagonal of the axial section is 34 cm, what are the radius and height of the cylinder?

Work through these problems, and you'll become a cylinder calculation pro in no time!

So, there you have it! We've successfully navigated the world of cylinders, proportions, and the Pythagorean theorem. Remember, math can be fun if you approach it with a clear strategy and a willingness to break down problems into smaller, digestible parts. Keep practicing, keep exploring, and most importantly, keep learning! Good luck, and happy calculating!