Problem 90 - Intermediate Algebra

Question

Hockey. A goal is scored in hockey when the puck, a vulcanized rubber disk \(2.5 \mathrm{cm}(1 \text { in. ) thick and } 7.6 \mathrm{cm}(3 \mathrm{in.})\) in diameter, is driven into the opponent's goal. Find the volume of a puck in cubic centimeters and cubic inches. Round to the nearest tenth.

Step-by-Step Solution

Answer

The puck's volume is approximately 113.1 cm³ and 7.1 in³.

Step 1: Understand the Formula for Volume of a Cylinder

The volume of a cylinder is calculated using the formula: \( V = \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height of the cylinder.

Step 2: Convert Diameter to Radius

The diameter of the puck is given as \(7.6\, \text{cm}\) or \(3\, \text{inches}\). To find the radius, divide the diameter by 2. For centimeters, \( r = \frac{7.6}{2} = 3.8 \) cm. For inches, \( r = \frac{3}{2} = 1.5 \) inches.

Step 3: Plug Values into the Formula (Centimeters)

Using the radius \( r = 3.8\, \text{cm} \) and height \( h = 2.5\, \text{cm} \), substitute into the volume formula: \( V = \pi (3.8)^2 (2.5) \). Calculate the volume: \( V \approx 3.14159 \times 14.44 \times 2.5 \approx 113.1 \, \text{cm}^3 \).

Step 4: Plug Values into the Formula (Inches)

Using the radius \( r = 1.5\, \text{in} \) and height \( h = 1\, \text{in} \), substitute into the volume formula: \( V = \pi (1.5)^2 (1) \). Calculate the volume: \( V \approx 3.14159 \times 2.25 \approx 7.1 \, \text{in}^3 \).

Step 5: Round the Results

Round the calculated volumes to the nearest tenth: The volume of the puck is approximately \(113.1\, \text{cm}^3\) and \(7.1\, \text{in}^3\).

Key Concepts

Understanding Geometry in Cylinders

In the world of geometry, a cylinder is a three-dimensional shape with two parallel circular bases connected by a cylindrical surface. Understanding the basic properties of a cylinder is key to solving many volume-related problems.

A cylinder has:

• Height: The perpendicular distance between the two bases
• Radius: The distance from the center of the base to its edge
• Diameter: Twice the radius, which runs across the circular base

To solve problems involving the cylinder, it's crucial to first understand these properties, which will help you correctly apply the cylindrical volume formula later.

Using the Cylindrical Volume Formula

The cylindrical volume formula is essential for calculating the capacity of cylinders. To find the volume, use the formula: \[ V = pi r^2 h \]
Here, \( \pi \) is about 3.14159, \( r \) is the radius, and \( h \) is the height. For a hockey puck, which is a small cylinder, the radius is half of its diameter. You plug these values into the formula to calculate how much space the puck occupies.

To illustrate:

• For a puck with a diameter of 7.6 cm and height of 2.5 cm, the volume calculation would go as follows: find the radius by dividing the diameter by two (3.8 cm), then compute the volume using \( V = pi (3.8)^2 (2.5) \).
• This results in approximately 113.1 cubic centimeters.

Understanding and applying this formula is fundamental to many topics in geometry, especially when dealing with any cylindrical shape.

The Importance of Unit Conversion

Unit conversion is a critical skill in math and science, ensuring you calculate correctly when measurements are given in different units. Often in geometry problems, and particularly in calculating volumes, you may need to convert between measurement units like centimeters and inches.

Here’s how it works:

• First, know the conversion factors: 1 inch equals 2.54 centimeters. This means if you have a diameter in inches (3 inches) and need to convert it to centimeters, multiply by 2.54 to get 7.6 cm.
• Similarly, to convert from centimeters to inches, you divide by 2.54.

Understanding unit conversion helps keep your calculations accurate and ensures that when you calculate the volume of objects like a puck, the units are consistent, making the results reliable. These skills are invaluable for dealing with real-world problems where precise measurement and unit management are crucial.