Problem 71 - Algebra A Combined Function

Question

The function \(A(r)=\pi r^{2}\) may be used to find the area of a circle if we are given its radius. Use this function to answer Exercises 71 and 72. Find the area of a circle whose radius is 5 centimeters. (Do not approximate \(\pi .)\)

Step-by-Step Solution

Answer

The area is \( 25\pi \) square centimeters.

Step 1: Understanding the Problem

We are asked to find the area of a circle and are given the radius. We will use the formula for the area of a circle, which is given by the function \( A(r) = \pi r^2 \). Here, \( r \) is the radius of the circle.

Step 2: Substitute the Value

Given the radius \( r = 5 \) centimeters, substitute it into the formula: \( A(r) = \pi (5)^2 \).

Step 3: Calculate the Expression

Calculate the square of the radius: \((5)^2 = 25\). Substitute this back into the formula to get \( A(5) = \pi \times 25 \).

Step 4: Express Final Answer

The area of the circle is \( 25\pi \) square centimeters. We do not approximate \( \pi \), so leave the answer in terms of \( \pi \).

Key Concepts

Circle Formula

The circle formula is a cornerstone in geometry, helping us understand the relationship between a circle's radius and its area. In this case, the formula is expressed as a function, written as \( A(r) = \pi r^2 \). This simple yet powerful formula tells us that the area \( A \) of a circle can be calculated if we know the radius \( r \).
- The symbol \( \pi \) (pi) is a constant, approximately 3.14159, representing the ratio of a circle's circumference to its diameter.
- \( r \) stands for the radius, which is the distance from the center of the circle to any point on its edge. Using this formula, you can easily find the area of any circle by squaring the radius and multiplying by \( \pi \). This approach links the core elements of a circle into a single mathematical sentence.

Radius

The radius is a key component in many circle-related calculations. It is the distance from the center point of the circle to any point along its edge. Imagine it as the spoke of a wheel—it is crucial for determining the size of the circle. In our exercise, the radius \( r \) is given as 5 centimeters.
The significance of the radius in circle geometry includes:

• It influences the size of the circle: larger radii mean bigger circles.
• It is used to calculate both the area and the circumference.
• All radii in a single circle have the same length.

Knowing the radius allows us to use formulas like \( A(r) = \pi r^2 \) to find other geometric properties, such as the area. Hence, understanding and determining the radius correctly is fundamental in solving problems related to circles.

Mathematical Expression

A mathematical expression is a combination of numbers and symbols that represent a mathematical concept or relationship. In the context of our circle problem, the expression \( \pi r^2 \) represents the formula for the area of a circle.
- **\( \pi \)** is a symbol that stands for a mathematical constant.- **\( r^2 \)** indicates that you multiply the radius by itself, which is called squaring the radius.Together, they form the expression that shows how the radius and the constant \( \pi \) work together to calculate the area. By substituting the radius value into this expression, you can solve for specific numerical values, which is a common and crucial step in tackling math problems.

Function

Functions play a critical role in mathematics, allowing us to understand how inputs are related to outputs. In our problem, the function \( A(r) = \pi r^2 \) describes how the input \( r \) (the radius) determines the output \( A(r) \) (the area).
- This function is **dependent**. Changes in the radius will directly affect the area.- Functions like this simplify complex relationships into manageable equations.
By providing a clear rule (multiply the square of the radius by \( \pi \)), functions make it easier to predict and verify outcomes. They help us to not only calculate specific instances but also understand and describe general relationships between variables. Understanding functions and how they operate simplifies the problem-solving process, especially in geometry.