Problem 20 - Algebra A Combined Function

Question

The area of a circle whose radius is 9 meters is \((9 \cdot 9 \cdot \pi)\) square meters. Write this area using exponential notation.

Step-by-Step Solution

Answer

The area is \( 9^2\pi \) square meters.

Step 1: Read the Problem Statement

We need to express the area of a circle, which is given in terms of the radius and pi, in exponential notation.

Step 2: Understand the Area Formula for a Circle

The area of a circle is calculated using the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle and \( \pi \) is a mathematical constant.

Step 3: Plug the Radius into the Formula

Substitute the radius of 9 meters into the formula: \( A = \pi \times 9^2 \).

Step 4: Calculate the Area

Find the area by evaluating \( 9^2 \). This equals \( 81 \), so \( A = 81\pi \) square meters.

Step 5: Express the Result in Exponential Notation

Note that \( 9^2 \) can be expressed as \( 9^2 \). Therefore, the area in exponential notation is \( A = 9^2 \pi \).

Key Concepts

Area of a Circle

The area of a circle is a way to measure the space inside the boundary of a circle. It tells us how much flat, two-dimensional space is enclosed by the circle. This concept is important because circles are common in geometry, and calculating this space can be crucial for both theoretical and practical applications, like designing circular gardens or round tables. To find the area of a circle, we use a simple formula: \[A = \pi r^2\]Here, \( A \) represents the area, \( \pi \) is a mathematical constant, and \( r \) is the radius of the circle. This formula shows us how the area is related to the square of the radius and the constant \( \pi \), making it straightforward to calculate the circle's area when the radius is known. Remember, the unit for the area will always be squared, like square meters (\( m^2 \)), because it covers a two-dimensional space.

Radius

The radius of a circle is a fundamental part of understanding geometry and the properties of circles. It is the distance from the center of the circle to any point on its perimeter. You can think of the radius as a line segment that starts at the circle's center and stops at its edge. The radius is crucial because it directly influences the circle's size and is used in calculating both the circumference and the area of a circle. For example:

• In the area formula \( A = \pi r^2 \), the radius \( r \) is squared and multiplied by \( \pi \) to find the area.
• The larger the radius, the more space the circle encloses, thus a bigger area.

Knowing the radius can help solve many problems, whether you're working on a math test or figuring out the length of a circular track.

Pi

Pi, commonly written as \( \pi \), is one of the most remarkable and well-known mathematical constants. Pi represents the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159. This ratio remains constant for all circles, regardless of their size, making \( \pi \) a fundamental element in geometry. Pi plays a key role in many formulas, especially those dealing with circles:

• In the area formula \( A = \pi r^2 \), \( \pi \) is multiplied by the square of the radius \( r \)
• In the circumference formula \( C = 2\pi r \), it helps calculate the perimeter of the circle.

Pi is an irrational number, meaning it cannot be exactly expressed as a fraction, and its decimal representation never ends or repeats. Still, for practical purposes, it's often approximated by 3.14 or \( \frac{22}{7} \). Pi's presence in geometry and mathematics makes it a cornerstone of calculations involving circles.

Mathematical Constant

A mathematical constant is a special number that arises naturally in mathematics and can be used to solve various problems across many fields. Mathematical constants like pi (\( \pi \)) or Euler's number \( e \) have unique and universal values. These constants are crucial not only for computations but also for understanding fundamental properties and relationships in mathematical theories. Mathematical constants are:

• Fixed Value: They don’t change after being defined, which makes them reliable tools in calculations.
• Non-changing Nature: They remain the same across different contexts and applications, providing a universal tool for mathematicians.

For circles, pi (\( \pi \)) is a key constant representing the constant ratio between the circumference and diameter, helping students and professionals alike in diverse calculations. Understanding these constants enriches one's ability to work with mathematical concepts efficiently.