Problem 15 - Basic Technical Mathematics with Calculus

Question

Find the area of the circle with the given radius or diameter. $$d=2.33 \mathrm{m}$$

Step-by-Step Solution

Answer

The area of the circle is approximately 4.26499 square meters.

Step 1: Understand the formula for the area of a circle

To find the area of a circle, we use the formula: \( A = \pi r^2 \), where \( A \) represents the area and \( r \) is the radius of the circle.

Step 2: Convert diameter to radius

The diameter \( d \) is given as \( 2.33 \) meters. The radius \( r \) is half of the diameter. Calculate the radius: \( r = \frac{d}{2} = \frac{2.33}{2} = 1.165 \text{ meters} \).

Step 3: Apply the radius in the area formula

Substitute the radius \( r = 1.165 \) meters into the area formula \( A = \pi r^2 \). This gives us: \( A = \pi (1.165)^2 \).

Step 4: Calculate the area using the formula

First, calculate \( (1.165)^2 = 1.357225 \). Then, multiply by \( \pi \) to find the area: \( A = \pi \times 1.357225 \approx 4.26499 \text{ square meters} \).

Key Concepts

Circle Geometry

Circle geometry is a fundamental aspect of mathematics that explores the properties and dimensions of circles. A circle is a shape with all points equidistant from a center point. The main components of a circle include the radius, diameter, and circumference.
The radius is the distance from the center to any point on the circle, while the diameter is twice the radius and spans from one point on the circle, through the center, to another point on the edge. Understanding these basic elements is crucial for solving problems involving circles, such as calculating area or circumference. The formula for the circumference is usually given as \( C = \pi d \) or \( C = 2\pi r \), and the area is found using \( A = \pi r^2 \). Grasping these concepts allows students to tackle various mathematical challenges involving circles with confidence.

Diameter to Radius Conversion

Converting a diameter to a radius is a straightforward yet essential skill in circle geometry. The radius is simply half the diameter.

• For example, if you have the diameter \( d = 2.33 \) meters, the radius is calculated by dividing the diameter by 2.
• This means \( r = \frac{d}{2} = \frac{2.33}{2} = 1.165 \) meters.

By understanding this conversion, it becomes easier to use other formulas that require the radius, such as calculating the area or the circumference of a circle. Converting these measurements allows more flexibility and understanding in solving real-world problems involving circles.

Mathematical Formulas

Mathematical formulas are powerful tools that allow us to compute complex values in a structured manner. They serve as the recipe for solving mathematical problems, like finding the area of a circle.
To find the area of a circle, you use the formula \( A = \pi r^2 \). In this equation, \( \pi \) is a constant approximately equal to 3.14159, and \( r \) is the radius of the circle. By correctly inputting the values into this formula, you can calculate the area with precision.
Formulas like these are crucial in applied mathematics, translating real-world scenarios into mathematical expressions that can be solved systematically.

Applied Mathematics

Applied mathematics involves using mathematical methods and formulas to solve practical problems. Calculating the area of a circle is a prime example of applied mathematics, where you take a real-world radius or diameter and compute a tangible measurement.
In many fields, such as engineering, architecture, and various sciences, understanding these calculations can have practical implications. For example, knowing the area of a circle can help in determining the amount of material needed to cover a circular surface or in designing circular components in mechanical systems.

• By mastering circle geometry and its related formulas, one can solve complex problems that appear in real-life situations.
• This skill is invaluable for both academic success and practical applications, bridging the gap between abstract mathematical concepts and real-world problem solving.

Applied mathematics thus empowers individuals to leverage mathematics as a universal language to understand and shape the world around us.