Problem 2 - Technical Mathematics with Calculus

Question

Find the volume and the lateral area of a right circular cylinder having a base radius of 128 and a height of 285.

Step-by-Step Solution

Answer

The volume of the cylinder is approximately \(\pi \times 128^2 \times 285\) cubic units and the lateral area is approximately \(2\pi \times 128 \times 285\) square units.

Step 1: Calculate the Volume

The volume of a right circular cylinder can be calculated using the formula: Volume = \(\pi r^2 h\), where \(\pi\) is a constant (approximately 3.14159), \(r\) is the radius of the cylinder's base, and \(h\) is the height of the cylinder. Plug the values \(r = 128\) and \(h = 285\) into the formula to get the volume.

Step 2: Calculate the Lateral Area

The lateral area of a right circular cylinder can be found using the formula: Lateral Area = \(2\pi r h\), where \(r\) is the radius and \(h\) is the height of the cylinder. Use the provided values \(r = 128\) and \(h = 285\) to calculate the lateral area.

Step 3: Solve for the Volume

Substituting the given numbers into the volume formula yields Volume = \(\pi (128)^2 \cdot 285\) cubic units. Calculate this to obtain the final volume.

Step 4: Solve for the Lateral Area

Substituting into the lateral area formula gives Lateral Area = \(2\pi \cdot 128 \cdot 285\) square units. Compute this to obtain the final lateral area.

Key Concepts

Cylinder Volume Calculation

Understanding the volume of a right circular cylinder is crucial in various fields, including engineering and architecture. The formula to calculate this volume is relatively straightforward:

Volume = \( \pi r^2 h \),
where \( \pi \) represents a mathematical constant approximately equal to 3.14159, \( r \) is the radius of the cylinder's base, and \( h \) is the height of the cylinder.
To find the volume of our cylinder with a base radius of 128 units and height of 285 units, we substitute these values into the formula to get:
\( \pi \times 128^2 \times 285 \) cubic units. By performing this operation, we get the three-dimensional space occupied by the cylinder.

Lateral Area of a Cylinder

The lateral area is the surface area of the sides of a cylinder, excluding the top and bottom faces. The formula to calculate the lateral area of a right circular cylinder is:

Lateral Area = \( 2\pi rh \),
where \( \pi \) again is the mathematical constant pi, \( r \) is the radius, and \( h \) is the height.
For our cylinder, we would use the given radius of 128 units and height of 285 units. Substituting those into our formula reveals:
\( 2\pi \times 128 \times 285 \) square units as the lateral area. This value represents the total area of the cylinder's curved surface.

Mathematical Constants

In geometry, a constant is a value that does not change. The most familiar constant we engage with in the context of cylinders is \( \pi \), the ratio of a circle's circumference to its diameter. It's an irrational number, meaning its exact value is impossible to determine, and it continues infinitely without repeating. In many calculations, we approximate \( \pi \) to 3.14159, but when precision is key, we may use more digits such as 3.14159265359 or rely on a calculator's built-in \( \pi \) function. This constant is fundamental not only in circular geometry but in many areas of mathematics, science, and engineering.

Geometry in Technical Mathematics

Geometry plays a vital role in technical mathematics, a branch of mathematics that applies mathematical theory and computation to solve real-world problems with a high level of precision. Right circular cylinders, for example, are geometric shapes commonly encountered in technical design and calculations. They are defined by parameters such as radius, height, and the relationships between these dimensions, captured by exact formulas for volume and surface area. Understanding these relationships is key to practical applications such as determining the capacity of containers or the material needed to create cylindrical objects.