Cylinder B Height Calculation Given Volume And Cylinder A Radius

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In the realm of geometry, cylinders stand as fundamental three-dimensional shapes. Understanding their properties, particularly volume and dimensions, is crucial for various applications. This article delves into a problem involving two cylinders, Cylinder A and Cylinder B. We are given the radius of Cylinder A and the volume of Cylinder B. Our mission is to determine the height of Cylinder B. This exploration will not only reinforce our understanding of cylinder geometry but also highlight the practical application of mathematical formulas in solving real-world problems.

Understanding Cylinder Geometry

Before we dive into the problem, let's solidify our understanding of cylinders and their properties. A cylinder, in its simplest form, is a three-dimensional geometric shape with two parallel circular bases connected by a curved surface. Key parameters that define a cylinder are its radius (r) and height (h). The radius is the distance from the center of the circular base to any point on its circumference, while the height is the perpendicular distance between the two bases.

The volume (V) of a cylinder is the amount of space it occupies and is calculated using the formula:

V = π r² h

Where:

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

This formula is a cornerstone in solving problems related to cylinders, including the one we are about to tackle. By rearranging the formula, we can also find the radius or height if the volume and one of these dimensions are known. This is precisely the approach we will use to determine the height of Cylinder B.

Problem Statement

Let's restate the problem at hand. We have two cylinders, Cylinder A and Cylinder B. Cylinder A has a radius of 4 centimeters. Cylinder B has a volume of 176π cubic centimeters. The core question we aim to answer is: What is the height of Cylinder B?

This problem provides us with specific information about each cylinder. For Cylinder A, we know its radius, which is a crucial dimension. For Cylinder B, we are given its volume, a measure of the space it occupies. However, we lack information about Cylinder B's radius or height. To find the height, we will need to utilize the formula for the volume of a cylinder and apply some algebraic manipulation.

Solution Approach

To solve this problem, we will employ a step-by-step approach, leveraging our understanding of cylinder geometry and algebraic principles. Here's the plan:

  1. Write down the formula for the volume of a cylinder: V = π r² h
  2. Identify the knowns and unknowns:
    • For Cylinder B, we know the volume (V = 176π cubic centimeters) and need to find the height (h).
    • We don't know the radius of Cylinder B, so we'll represent it as r.
  3. Substitute the known values into the formula: 176π = π r² h
  4. Simplify the equation: Notice that π appears on both sides of the equation. We can divide both sides by π to simplify, resulting in: 176 = r² h
  5. Recognize the missing information: We have one equation but two unknowns (r and h). This means we cannot directly solve for h without additional information.
  6. Make a crucial assumption: Since the problem does not provide the radius of Cylinder B, let's assume that Cylinder B has the same radius as Cylinder A. This is a critical step that allows us to proceed with the solution. It's important to note that without this assumption, the problem would be unsolvable with the given information.
  7. Substitute the radius of Cylinder A into the equation: Since we're assuming the radius of Cylinder B is the same as Cylinder A, we have r = 4 centimeters. Substitute this into the equation: 176 = 4² h
  8. Solve for h:
    • Simplify: 176 = 16 h
    • Divide both sides by 16: h = 176 / 16
    • Calculate: h = 11 centimeters

Therefore, based on the assumption that Cylinder B has the same radius as Cylinder A, the height of Cylinder B is 11 centimeters.

Detailed Solution with Step-by-Step Calculation

Let's walk through the solution with a detailed breakdown of each step:

  1. Volume Formula: We begin with the fundamental formula for the volume of a cylinder:

    V = π r² h

    This formula relates the volume (V) to the radius (r) and height (h) of the cylinder.

  2. Knowns and Unknowns: For Cylinder B, we have:

    • Volume (V) = 176π cubic centimeters
    • Radius (r) = Unknown
    • Height (h) = Unknown (This is what we want to find)

  3. Substitution: Substitute the known volume into the formula:

    176π = π r² h

    This equation now includes the specific volume of Cylinder B.

  4. Simplification: Divide both sides of the equation by π:

    176 = r² h

    This simplification eliminates π, making the equation easier to work with.

  5. Missing Information: We have one equation (176 = r² h) and two unknowns (r and h). This indicates that we need more information or an assumption to solve for h.

  6. Assumption: We assume that Cylinder B has the same radius as Cylinder A (which is 4 centimeters). This is a crucial step that allows us to proceed. If the radii were different, we wouldn't be able to solve for the height with the given information. By making this assumption, we add a constraint that allows us to find a solution.

  7. Substitute Radius: Substitute the radius of Cylinder A (r = 4 centimeters) into the equation:

    176 = 4² h

    This substitution replaces the unknown radius r with a known value.

  8. Solve for Height:

    • Simplify: 176 = 16 h
    • Divide both sides by 16: h = 176 / 16
    • Calculate: h = 11 centimeters

    This series of algebraic steps isolates h, allowing us to calculate its value.

Final Answer

Based on the assumption that Cylinder B has the same radius as Cylinder A, the height of Cylinder B is 11 centimeters. It is important to remember that this answer is contingent upon the assumption we made about the radii being equal. If the radii were different, the height of Cylinder B would also be different.

This problem has demonstrated how to calculate the height of a cylinder given its volume and the radius of another cylinder, under a crucial assumption. We've reinforced our understanding of the cylinder volume formula (V = π r² h) and the importance of identifying knowns, unknowns, and any necessary assumptions in problem-solving. The key takeaway here is not just the numerical answer, but the process of applying geometric principles and algebraic techniques to solve a practical problem. Furthermore, this exercise highlights the significance of clear problem statements and the potential for ambiguity when information is lacking. In real-world scenarios, it's crucial to gather all necessary information or make reasonable assumptions, clearly stating them, to arrive at a meaningful solution. The ability to analyze problems, identify the relevant formulas, and make logical deductions is a valuable skill in mathematics and beyond.