Finding The Height Of Cylinder B A Step-by-Step Solution

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In the realm of geometry, understanding the properties and relationships of three-dimensional shapes is paramount. Cylinders, with their elegant simplicity, are fundamental geometric figures that find applications in various fields, from engineering to architecture. In this comprehensive exploration, we embark on a journey to unravel the intricacies of cylinder calculations, specifically focusing on determining the height of a cylinder given its volume and the radius of another cylinder. This detailed guide will walk you through the problem-solving process, ensuring clarity and comprehension at every step.

At the heart of our exploration lies the cylinder, a geometric shape characterized by its two parallel circular bases connected by a curved surface. Imagine a can of soup or a drinking glass – these are everyday examples of cylinders. To fully grasp the concept of a cylinder, let's delve into its key properties:

  • Bases: Cylinders possess two congruent circular bases that lie in parallel planes. These bases define the cylinder's circular boundaries.
  • Radius: The radius of a cylinder is the distance from the center of its circular base to any point on the circumference. It essentially measures the "width" of the cylinder's circular end.
  • Height: The height of a cylinder is the perpendicular distance between its two bases. It represents the cylinder's "length" or "vertical extent."
  • Volume: The volume of a cylinder is the amount of space it occupies. It's a measure of the cylinder's three-dimensional capacity.

With a firm understanding of these properties, we can now proceed to the mathematical formula that governs a cylinder's volume.

The volume of a cylinder is calculated using a straightforward formula that elegantly connects its radius, height, and the mathematical constant pi (Ï€), which is approximately equal to 3.14159. The formula is expressed as follows:

Volume = π * radius² * height

This formula reveals a fundamental relationship: the volume of a cylinder is directly proportional to the square of its radius and its height. This means that if you double the radius, the volume quadruples (increases by a factor of four), and if you double the height, the volume doubles.

Now, let's apply this knowledge to the specific problem at hand: determining the height of cylinder B.

We are presented with a scenario involving two cylinders, cylinder A and cylinder B. Cylinder A's radius is given as 4 centimeters, while cylinder B's volume is known to be 176Ï€ cubic centimeters. Our mission is to calculate the height of cylinder B.

To tackle this problem, we will employ a systematic approach, leveraging the volume formula and the information provided. Let's break down the solution into manageable steps.

Before diving into calculations, it's crucial to identify what we already know and what we need to find. In this case:

  • Knowns:
    • Radius of cylinder A = 4 centimeters
    • Volume of cylinder B = 176Ï€ cubic centimeters
  • Unknown:
    • Height of cylinder B

With a clear understanding of the givens and the goal, we can proceed to the next step.

Since we're interested in the height of cylinder B, we'll focus on the volume formula specifically for cylinder B. Let's denote the radius of cylinder B as rB and its height as hB. Then, the volume formula for cylinder B becomes:

Volume of cylinder B = π * rB² * hB

We know the volume of cylinder B (176Ï€ cubic centimeters), but we don't know its radius (rB). However, we do have information about cylinder A's radius, which we might be able to use indirectly.

At this point, you might be wondering how the radius of cylinder A (4 centimeters) comes into play. The problem doesn't explicitly state any relationship between the radii of the two cylinders. However, it's crucial to carefully analyze the problem statement. Notice that the problem asks for the height of cylinder B given its volume and the radius of 4 centimeters. This suggests that the 4 centimeters is likely related to cylinder B, and not cylinder A.

Therefore, we can reasonably assume that the radius of cylinder B (rB) is equal to the radius given, which is 4 centimeters.

Now that we have the radius of cylinder B (rB = 4 centimeters) and its volume (176Ï€ cubic centimeters), we can substitute these values into the volume formula and solve for the height (hB):

176π = π * (4)² * hB

Simplifying the equation:

176π = π * 16 * hB

Divide both sides by 16Ï€:

hB = 176Ï€ / (16Ï€)

The π terms cancel out:

hB = 176 / 16

Perform the division:

hB = 11 centimeters

Therefore, the height of cylinder B is 11 centimeters.

Now that we know the height of cylinder B, let’s verify by plugging the known and newly found values in the cylinder volume formula. This step will ensure that our solution is accurate.

Volume of cylinder B = π * (4 cm)² * 11 cm Volume of cylinder B = π * 16 cm² * 11 cm Volume of cylinder B = 176π cm³

The volume we calculated matches the volume given in the problem. So, our answer is accurate.

We have successfully determined the height of cylinder B by systematically applying the volume formula and leveraging the given information. This problem-solving journey highlights the importance of understanding geometric principles, careful analysis, and step-by-step execution. By mastering these skills, you can confidently tackle a wide range of geometric challenges.

  • The volume of a cylinder is calculated using the formula: Volume = Ï€ * radius² * height.
  • Carefully analyze the problem statement to identify the knowns and unknowns.
  • Strategically utilize the given information to make logical deductions.
  • Substitute known values into the appropriate formula and solve for the unknown.
  • Always double-check your answer to make sure it's accurate.

To solidify your understanding, try solving these practice problems:

  1. A cylinder has a radius of 5 centimeters and a volume of 250Ï€ cubic centimeters. What is its height?
  2. A cylinder has a height of 8 centimeters and a volume of 128Ï€ cubic centimeters. What is its radius?
  3. Cylinder X has a radius of 3 centimeters and a height of 10 centimeters. Cylinder Y has a radius of 6 centimeters. If cylinder X and cylinder Y have the same volume, what is the height of cylinder Y?

By diligently practicing and applying the concepts learned in this guide, you'll enhance your problem-solving abilities and gain a deeper appreciation for the elegance of geometry.