Problem 48 - College Algebra

Question

The volume formula for a cylinder is \(V=\pi r^{2} h\) Using the symbol \(\pi\) in your answer, find the volume of a cylinder with a radius, \(r,\) of 4 \(\mathrm{cm}\) and a height of 14 \(\mathrm{cm} .\)

Step-by-Step Solution

Answer

The volume of the cylinder is \( 224\pi \) cubic centimeters.

Step 1: Identify the Variables

We are given the radius \( r = 4 \) cm and the height \( h = 14 \) cm. We will use these values to substitute into the volume formula for a cylinder.

Step 2: Write the Formula for Volume

The formula for the volume \( V \) of a cylinder is given by \( V = \pi r^2 h \).

Step 3: Substitute Values into the Formula

Replace \( r \) with 4 cm and \( h \) with 14 cm in the formula, so it becomes \( V = \pi (4)^2 (14) \).

Step 4: Calculate \( r^2 \)

Calculate \( 4^2 \), which gives us 16.

Step 5: Multiply by Height

Now, multiply 16 by 14 (which is the height): \( 16 \times 14 = 224 \).

Step 6: Include \( \pi \) in the Result

Combine all parts from previous steps into the final formula: \( V = 224\pi \). This represents the volume of the cylinder in cubic centimeters.

Key Concepts

Cylinder

A cylinder is a three-dimensional geometric shape with two parallel circular bases connected by a curved surface. Imagine a can; it forms a similar shape. The circle at the top has a counterpart at the bottom, and they're aligned. The distance between these two circles is the height of the cylinder.

Cylinders are everywhere – from soup cans to tubes. Understanding their structure is key.

• Two bases: These circular parts are equal in size.
• Curved surface: This wraps around the two bases.
• Height: The length from one base to the other.
• Radius: The distance from the center of the base to its edge.

Feel free to visualize it as a rolled-up rectangle! Knowing the parts of a cylinder makes calculations like volume easier.

Mathematical Formula

One essential formula related to cylinders calculates their volume, which is the number of cubic units needed to fill the cylinder. The formula is:

\[ V = \ \pi r^2 h \]

Here's a breakdown:

• \( V \) stands for volume.
• \( \pi \) is a mathematical constant approximately equal to 3.14159. It's the ratio of a circle's circumference to its diameter.
• \( r \) is the radius of the base.
• \( h \) is the height of the cylinder.

This formula integrates basic mathematical concepts, allowing you to use the cylinder's dimensions to find its volume.

Problem-Solving Steps

When faced with an exercise involving the volume of a cylinder, follow a series of methodical steps to ensure accuracy. Let's break it down:

• Identify Variables: Gather the values for radius \( r \) and height \( h \) provided in the problem.
• Apply the Formula: Use the equation \( V = \pi r^2 h \) to find the volume.
• Substitute Values: Plug \( r = 4 \) cm and \( h = 14 \) cm to formulate \( V = \pi (4)^2 (14) \).
• Calculate Square: Work out \( 4^2 \) which equals 16.
• Multiply by Height and Pi: Multiply 16 by the height, then incorporate \( \pi \). The final expression becomes \( V = 224\pi \).

Following these steps ensures that you systematically solve for the cylinder's volume.

Geometry

Geometry is a branch of mathematics that deals with the properties and relations of points, lines, surfaces, and solids. Cylinders fall under geometry because they are solid figures. Understanding basic geometry helps identify the three-dimensional nature of such shapes.

Key elements of geometry when studying cylinders include:

• Understanding Circles: As a cylinder has circular bases, it's crucial to grasp concepts like radius and diameter.
• Spatial Awareness: Conceptualizing the height and how it relates to other parts.
• Volume Calculation: Using measurements of radius and height, helps us find space inside the shape.

Recognizing these aspects fosters a deeper appreciation for geometry and its applications like calculating cylinder volumes.

Mathematics Education

Mathematics education equips students with the crucial skills needed to tackle problems involving shapes like cylinders. Learning concepts such as volume of a cylinder helps build a strong foundation in geometry and basic math.

By understanding how to calculate the volume, students:

• Enhance logical reasoning and problem-solving abilities.
• Gain practical skills applicable in real-world situations like engineering and design.
• Develop a deeper understanding of three-dimensional geometry.

Teaching and learning these concepts can be both engaging and rewarding, providing students with tools they need throughout their educational journey.