Cylinder Volume Change: Radius Halved Explained

Have you ever wondered what happens to the volume of a cylinder if you change its dimensions? Let's dive into a common mathematical scenario: creating a cylinder with specific dimensions and exploring how its volume changes when we alter its radius. This article will walk you through the process step-by-step, making it easy to grasp the concepts and calculations involved. We'll use a cylinder with a height of 14 cm and a radius of 10 cm as our starting point and then investigate what happens when we halve the radius. This is a fundamental concept in geometry and has practical applications in various fields, from engineering to everyday problem-solving. Understanding these principles helps us predict and control outcomes when dealing with three-dimensional shapes.

Initial Cylinder Volume Calculation

To begin, let's calculate the volume of our initial cylinder. The formula for the volume (V) of a cylinder is given by V = πr²h, where r is the radius and h is the height. In our case, the height (h) is 14 cm, and the radius (r) is 10 cm. Plugging these values into the formula, we get:

V = π * (10 cm)² * 14 cm V = π * 100 cm² * 14 cm V = 1400π cm³

So, the volume of the original cylinder is 1400π cubic centimeters. This initial calculation sets the baseline for our comparison. We now have a concrete value to which we can compare the new volume when the radius is changed. It's crucial to remember this initial volume as it will help us understand the proportional change later. The value of π is approximately 3.14159, but we'll keep it in terms of π for precise calculations and easier comparison. This approach highlights the elegance of mathematical expressions and how they can simplify complex problems.

Visualizing the Impact of Radius on Volume

Imagine filling the original cylinder with water. Its capacity is 1400π cubic centimeters. Now, think about what happens to this capacity when we change the radius. The radius is squared in the volume formula, which means changes to the radius will have a more significant impact on the volume than changes to the height. This is a critical insight for understanding the relationship between a cylinder's dimensions and its volume. The volume is directly proportional to the square of the radius, meaning if you double the radius, you quadruple the volume (assuming the height remains constant). This relationship is not intuitive for everyone, which is why visualizing it and understanding the formula is essential.

Halving the Radius: New Volume Calculation

Now, let's halve the radius of the cylinder. Our new radius (r') will be 10 cm / 2 = 5 cm. The height remains the same at 14 cm. We'll use the same formula for the volume, but with the new radius:

V' = π * (5 cm)² * 14 cm V' = π * 25 cm² * 14 cm V' = 350π cm³

The new volume (V') is 350π cubic centimeters. This is significantly less than the original volume. But how much less? To find out, we'll compare the new volume to the original volume. This step is crucial for understanding the proportional change that occurred when the radius was halved. The comparison will also help us identify the correct answer from the options provided. Proportional changes are fundamental in mathematics and physics, helping us understand how variables relate to each other.

Understanding the Proportional Change

To understand the proportional change, we will divide the new volume by the original volume:

V' / V = (350π cm³) / (1400π cm³) V' / V = 350 / 1400 V' / V = 1 / 4

This calculation shows that the new volume is 1/4 of the original volume. When the radius is halved, the volume becomes one-quarter of its initial value. This is because the radius is squared in the volume formula. Halving the radius reduces the radius squared by a factor of four (since (1/2)² = 1/4). This principle is crucial for understanding how changes in dimensions affect volume in three-dimensional shapes. It also illustrates the power of mathematical relationships in predicting outcomes.

Conclusion: The Impact of Halving the Radius

In conclusion, when the radius of a cylinder is halved, the volume becomes 1/4 of the original volume. This is due to the quadratic relationship between the radius and the volume in the formula V = πr²h. This understanding has significant implications in various real-world scenarios. For example, in engineering, this principle is used to design containers and tanks, ensuring they have the desired capacity. In cooking, understanding volume changes helps in adjusting recipes when using different sized pans or containers.

Therefore, the correct answer is:

A. The volume is 1/4 of the original.

This exploration demonstrates how a simple change in one dimension can significantly impact the volume of a three-dimensional object. By understanding the mathematical relationships involved, we can make informed decisions and solve practical problems more effectively. Mathematics is not just about formulas; it's about understanding the world around us. To further enhance your understanding of cylinder volumes and related concepts, you might find helpful resources on websites like Khan Academy's Geometry Section.

Further Exploration and Real-World Applications

This principle extends beyond just cylinders. The relationship between dimensions and volume is a core concept in understanding other geometric shapes as well. For example, in spheres, the volume is proportional to the cube of the radius, meaning changes in the radius have an even more dramatic impact on the volume. Exploring these relationships helps build a strong foundation in geometry and spatial reasoning. Real-world applications include everything from designing storage containers to understanding fluid dynamics. Consider how this principle might apply in situations like designing a fuel tank for a vehicle or calculating the amount of liquid a cylindrical pipe can carry. These are practical scenarios where a solid grasp of geometric principles is essential.