Problem 18 - Applied Physics

Question

Find the volume of a tractor engine cylinder whose radius is \(3.90 \mathrm{~cm}\) and whose length is \(8.00 \mathrm{~cm}\).

Step-by-Step Solution

Answer

The volume of the cylinder is approximately 382.68 cm³.

Step 1: Understanding the Formula

The formula for finding the volume of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height or length of the cylinder. In this problem, \( r = 3.90 \text{ cm} \) and \( h = 8.00 \text{ cm} \).

Step 2: Substituting the Values

Insert the given values of the radius and the height into the formula. Thus, the volume \( V \) is calculated as \( V = \pi (3.90)^2 (8.00) \).

Step 3: Calculating Step-by-Step

First, calculate the square of the radius: \( (3.90)^2 = 15.21 \). Then multiply this result by the height: \( 15.21 \times 8.00 = 121.68 \).

Step 4: Final Calculation

Finally, multiply the result from the previous step by \( \pi \) to find the volume: \( V = 121.68 \pi \approx 382.68 \, \text{cm}^3 \) (using \( \pi \approx 3.14159 \)).

Key Concepts

Understanding Cylinder Geometry

A cylinder is a three-dimensional shape with two parallel circular bases that are connected by a curved surface. It can be found in everyday objects like soda cans, pipes, and even some types of batteries. The main elements of a cylinder you need to familiarize yourself with include:

• The **radius (r)**: this is the distance from the center to the edge of the circular base.
• The **height (h)**: this is the distance between the two circular bases. It is sometimes referred to as the length of the cylinder.
• The **diameter**: this is twice the radius, the full width of the base circle.

In the context of our exercise, we're considering a "standing" cylinder, with the bases horizontal, just like a can standing upright.
Understanding these elements is crucial because they are used in the formula to calculate the volume of a cylinder, which measures how much space the cylinder occupies.

Volume Calculation of a Cylinder

Calculating the volume of a cylinder involves determining how much space is inside the cylinder. The general formula used to find the cylinder's volume is:
\[ V = \pi r^2 h \]Where:

• \( V \) is the volume,
• \( r \) is the radius of the circular base of the cylinder,
• \( h \) is the height (or length) of the cylinder.
• \( \pi \) (Pi) is a constant approximately equal to 3.14159.

This formula involves three main steps:1. **Squaring the radius**: Multiply the radius by itself. This part of the formula, \( r^2 \), determines the area of the circular base. 2. **Multiplying by the height**: This integrates the third dimension along the cylinder's length, giving a volume in cubic centimeters if the dimensions are in centimeters. 3. **Multiplying by \( \pi \)**: The factor of \( \pi \), which relates the circle's circumference to its diameter, completes the calculation for the volume.Applying this to our exercise, with a radius of 3.9 cm and a height of 8.0 cm, results in a volume of approximately 382.68 cubic centimeters. This is how much space the tractor engine cylinder would occupy.

Mathematical Problem Solving

Solving mathematical problems like finding the volume of a cylinder requires a combination of understanding, calculation, and verification. Here are some key tips to enhance your problem-solving skills in such scenarios:

• **Understand the Problem**: Identify what is given and what is being asked. In this case, you were given the dimensions of the cylinder and tasked with finding its volume.
• **Apply Formulas Correctly**: Ensure you are using the right formula and plugging in the values correctly. Errors here can lead to incorrect answers.
• **Perform Calculations Carefully**: Work through calculations methodically to avoid simple arithmetic errors, for example, in squaring the radius or multiplying by \( \pi \).
• **Verify Your Solution**: Always review your work. Double-check that all steps make sense and recalculate if necessary. This could involve repeating calculations, checking units, or reflecting on whether the answer seems reasonable given the context.

These strategies are pivotal in tackling any mathematical problem efficiently, instilling confidence that you can independently arrive at the correct solution.