Problem 2 - Basic Engineering Mathematics

Question

Find the volume and total surface area of a cylinder of length \(15 \mathrm{~cm}\) and diameter \(8 \mathrm{~cm}\).

Step-by-Step Solution

Answer

Volume: \(240\pi\, \text{cm}^3\); Surface Area: \(152\pi\, \text{cm}^2\).

Step 1: Understand the Given Values

We have a cylinder with a length (or height) of \(h = 15\) cm and a diameter \(d = 8\) cm. The radius \(r\) can be calculated as half of the diameter: \(r = \frac{d}{2} = 4\) cm.

Step 2: Calculate the Volume of the Cylinder

The volume \(V\) of a cylinder is given by the formula \(V = \pi r^2 h\). Substituting the known values: \(V = \pi (4)^2 (15) = \pi \times 16 \times 15 = 240\pi\, \text{cm}^3\).

Step 3: Calculate the Total Surface Area of the Cylinder

The total surface area \(A\) of a cylinder is given by the formula: \(A = 2\pi r (r + h)\). Substituting the known values: \(A = 2\pi (4)(4 + 15) = 2\pi \times 4 \times 19 = 152\pi\, \text{cm}^2\).

Step 4: Finalize the Results

The volume of the cylinder is \(240\pi\, \text{cm}^3\), and the total surface area is \(152\pi\, \text{cm}^2\). These values can be approximated using \(\pi \approx 3.14159\) if needed.

Key Concepts

Cylinder Surface Area

Understanding how to calculate the surface area of a cylinder is an important aspect of geometry. To find the total surface area, remember that a cylinder consists of two circle bases and one rectangular side that wraps around it.
For the circular bases, the area of each is found using the formula:

• Base area = \( \pi r^2 \) where \( r \) is the radius of the circle.

There are two bases, so their combined area is \(2\pi r^2\).
The side of the cylinder, when "unwrapped," is a rectangle. Its width is the height of the cylinder (\(h\)), and its length is the circumference of the base circle (\(2\pi r\)). Thus, the side's area is \(2\pi rh\).
To find the total surface area (\(A\)), add the areas of the two bases and the side:
\[ A = 2\pi r^2 + 2\pi rh = 2\pi r (r + h) \]

Geometry Formulas

In geometry, formulas like those used for cylinders allow us to compute characteristics like volume and surface area quickly. These formulas are derived from the basic properties of their shapes.
For example:

• The volume of a cylinder is calculated by taking the area of the base and multiplying it by the height, \( V = \pi r^2 h \).
• The surface area is derived by recognizing a cylinder as having two circular bases and a rectangular sheet, leading to its formula \( A = 2\pi r (r + h) \).

Memorizing key geometry formulas facilitates easier problem-solving in mathematics.
It's crucial to plug in the appropriate values for radius, diameter, and height to maintain accuracy when performing calculations.

Mathematical Calculations

Carrying out calculations for cylinder problems entails understanding
and using certain mathematical properties and constants, such as \( \pi \approx 3.14159 \).
The diameter of a cylinder is often provided, and finding the radius is essential since many formulas use it. Simply divide the diameter by two to get the radius (\( r = \frac{d}{2} \)).
Once you have all necessary measurements, substitute them directly into the formulas:

• Volume: \( V = \pi r^2 h \).
• Surface area: \( A = 2\pi r (r + h) \).

Apply these calculations carefully to ensure you're getting the right answers.
It's always a good practice to use approximate values for \( \pi \) when you need a numerical answer instead of leaving it in terms of \( \pi \). This helps in real-world applications where decimal precision may be needed.