Problem 2 - Applied Physics

Question

Find the volume of a cylinder whose height is \(7.50\) in. and diameter is \(4.20\) in. (Fig. 2.4).

Step-by-Step Solution

Answer

The volume of the cylinder is approximately 103.86 in³.

Step 1: Understand the Problem

The exercise asks us to find the volume of a cylinder. We are given the height of the cylinder as \( h = 7.50 \) in. and the diameter of its base as \( d = 4.20 \) in. To find the volume, we need both the height and the radius of the base.

Step 2: Calculate the Radius

The radius \( r \) is half of the diameter. Given the diameter \( d = 4.20 \) in., calculate the radius: \[r = \frac{d}{2} = \frac{4.20}{2} = 2.10 \text{ in.}\]

Step 3: Write the Formula for Volume

The formula for the volume \( V \) of a cylinder is given by: \[V = \pi r^2 h\] where \( r \) is the radius, and \( h \) is the height.

Step 4: Substitute the Known Values

Now, substitute the values of \( r \) and \( h \) into the volume formula:\[V = \pi (2.10)^2 (7.50)\]

Step 5: Calculate the Volume

First, calculate \( r^2 = (2.10)^2 = 4.41 \). Then calculate the volume:\[V = \pi \times 4.41 \times 7.50 = \pi \times 33.075\]Using \( \pi \approx 3.14159 \), the volume is:\[V \approx 3.14159 \times 33.075 \approx 103.86375 \text{ in}^3\]

Step 6: Round the Answer

Depending on the precision required, we can round the answer to a reasonable number of decimal places. If rounding to two decimal places, the volume is approximately \( 103.86 \) in³.

Key Concepts

Cylinder Geometry

Cylinders are three-dimensional shapes commonly seen in everyday life, like cans and tubes. Understanding their geometry is key to solving problems related to their volume and surface area. A cylinder consists of two parallel circular bases of the same size connected by a curved surface.
The distance between the bases is called the height of the cylinder. The radius is half the diameter of the base, and it’s crucial for volume calculations. Knowing these elements helps in applying mathematical formulas to find measures like volume.

Mathematics Problem Solving

Effective problem solving in mathematics involves a series of logical steps to find a solution. In this exercise, identifying known values like the height and diameter is the starting point. From there, calculating necessary dimensions like the radius helps simplify the process.
The radius is determined by halving the diameter, which is a critical step before using key formulas. Substituting these calculated values into known formulas, like the volume formula for a cylinder, is essential. Working systematically through these steps equips students to handle similar problems.

Surface Area and Volume

Surface area and volume are fundamental concepts in geometry involving measurements of a three-dimensional shape. While surface area concerns the total region covered by the surfaces of the cylinder, volume measures the space inside it.
For a cylinder, the volume is calculated using the formula \( V = \pi r^2 h \), where \( r \) is the radius of the base, and \( h \) is the height. It's a key aspect of understanding how much material can fit inside a cylinder or how much space it occupies. Correct computation is critical for practical applications.

Applied Calculations in Physics

In physics, calculations with cylinders have practical applications, such as determining capacities, fluid dynamics, and material strengths. Whether designing vessels or calculating forces, understanding how to compute a cylinder's volume is crucial.
For instance, when calculating the capacity of a container, knowing the volume allows us to determine how much material or fluid it can hold. These principles extend beyond simple calculations and are integral to engineering, manufacturing, and scientific research, underscoring the importance of mastering cylinder volume calculations.