Problem 19 - Basic Engineering Mathematics

Question

The area, \(A\), of a circle is given by \(A=\pi r^{2}\). Determine the area correct to 2 decimal places, given radius \(r=5.23 \mathrm{~m}\)

Step-by-Step Solution

Answer

The area of the circle is approximately 85.93 m².

Step 1: Identify the given values

We are given the radius of the circle as \( r = 5.23 \) meters. We need to find the area \( A \) of the circle using the formula \( A = \pi r^2 \).

Step 2: Substitute the radius into the area formula

Using the formula for the area of a circle, replace \( r \) with \( 5.23 \):\[ A = \pi (5.23)^2 \]

Step 3: Calculate \( r^2 \)

Compute \( 5.23^2 \):\[ 5.23^2 = 27.3529 \]

Step 4: Multiply by \( \pi \) to find the area

Substitute \( 27.3529 \) back into the area formula:\[ A = \pi \times 27.3529 \]Using the approximation \( \pi \approx 3.14159 \), calculate:\[ A \approx 3.14159 \times 27.3529 = 85.9258291 \]

Step 5: Round the area to two decimal places

Round 85.9258291 to two decimal places to get the final area:\[ A \approx 85.93 \text{ m}^2 \]

Key Concepts

Circle Formula

Understanding the formula for the area of a circle is a key part of many math problems. The formula is given by \( A = \pi r^2 \), where

• \( A \) is the area of the circle.
• \( \pi \) (pi) is a mathematical constant approximately equal to 3.14159.
• \( r \) is the radius of the circle, the distance from the center to the edge.

To find the area, you simply need to square the radius (multiply the radius by itself) and then multiply the result by \( \pi \). This formula "scales up" the simple notion of a square to work with the circular shape, where \( \pi \) accounts for the curve.
When solving problems, substituting the given radius into the formula is your first step.

Decimal Places

Decimal places are an essential feature in math for presenting precise values, especially when working with real-world measurements. When a number is rounded to "two decimal places," it is presented with two digits after the decimal point. This is often needed to ensure a consistent level of accuracy in results.
Consider the calculation of the area of a circle where the intermediate result is 85.9258291; rounding it to two decimal places gives you 85.93.

• Look at the third decimal place to round correctly.
• If it's 5 or more, increase the second decimal by one.
• If it's less than 5, leave the second decimal as is.

This approach helps maintain precision and readability without making the numbers overly complex.

Mathematical Constants

Mathematical constants like \( \pi \) are numbers that have a fixed value, independently of other variables. \( \pi \) is vital in formulas involving circles due to its role in defining the relationship between a circle's circumference and its diameter.
Unlike numbers we derive from calculations, constants come from fundamental math relationships. \( \pi \) represents the ratio of the circumference of any circle to its diameter and is approximately 3.14159.
Other common constants in math include Euler's number \( e \) and the square root of 2.

• Constants are often irrational, meaning they can't be completely expressed as a simple fraction.
• The decimals for these constants do not terminate, so we use approximations (like \( 3.14159 \) for \( \pi \)) in practical calculations.

Knowing these constants and how they apply is crucial for accurately solving mathematical problems.