Problem 52 - Algebra A Combined Function

Question

Given the circle below with radius \(5 y\) centimeters, find its area. Do not approximate \(\pi\).

Step-by-Step Solution

Answer

The area is \( 25\pi y^2 \) square centimeters.

Step 1: Recall the Formula for the Area of a Circle

The area of a circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle.

Step 2: Identify the Radius

The problem states that the radius of the circle is \(5y\) centimeters.

Step 3: Substitute Radius into the Area Formula

Substitute \( r = 5y \) into the area formula to get \( A = \pi (5y)^2 \).

Step 4: Simplify the Expression

Simplify \( (5y)^2 \) to obtain \( 25y^2 \). Thus, the area is \( A = \pi \times 25y^2 \).

Step 5: Write the Final Expression for the Area

The final expression for the area of the circle is \( A = 25\pi y^2 \) square centimeters.

Key Concepts

Circle Geometry

In the world of geometry, a circle is a simple yet fascinating two-dimensional shape. This shape is defined as the set of all points that are the same distance from a fixed point, known as the center. This distance is called the radius. Understanding circles involves several key components:

• Center: The fixed point from which every point on the circle is equidistant.
• Radius: The constant distance from the center to any point on the circle's edge.
• Diameter: A straight line passing through the center, connecting two points on the circle, twice as long as the radius.
• Circumference: The total distance around the circle.

These elements are fundamental when calculating properties like the area, which provides the interior space of the circle. And to find the area, we rely on specific mathematical formulas, which bring us to our next concept.

Mathematical Formula

A mathematical formula is a tool that helps us translate geometrical aspects into calculable numbers. For circles, one of the most essential formulas is used to find the area. This formula is expressed as:

\(A = \pi r^2\)

This formula tells us how the area is influenced by the size of the radius. Here are the elements explained:

• \(A\): Represents the area of the circle.
• \(\pi\): A mathematical constant approximately equal to 3.14159. It describes the ratio of the circumference to the diameter of any circle.
• \(r\): The radius of the circle.

This relationship shows why knowing just the radius enables us to calculate the entire area within the circle. Through substitution and simplification, we can easily compute the area with any given radius.

Algebraic Expression

Algebra allows us to manipulate symbols and numbers to represent quantities without specifying their exact values until necessary. In the case of finding the area of a circle, we encounter an algebraic expression. The expression from our problem, once the radius is given as \(5y\), becomes:

\(A = \pi (5y)^2\)

This expression can be simplified further:

• First, evaluate the square of the radius: \((5y)^2 = 25y^2\).
• Then substitute back into the formula: \(A = 25\pi y^2\).

Now, \(25\pi y^2\) denotes the area, which is an algebraic expression where the exact value depends on the variable \(y\). Understanding how to derive and simplify these expressions is crucial for solving many problems in mathematics, not just in geometry but across various disciplines.