Problem 72 - 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 circular garden whose radius is 8 feet. (Do not approximate \(\pi\).)

Step-by-Step Solution

Answer

The area of the garden is \(64\pi\) square feet.

Step 1: Understand the Problem

We need to find the area of a circular garden given its radius. The radius provided is 8 feet. We will use the formula for the area of a circle, which is given by the function \( A(r) = \pi r^2 \).

Step 2: Identify the Given Data

The radius \( r \) of the circular garden is given as 8 feet. This is the value we will substitute into our area formula.

Step 3: Substitute the Radius into the Area Formula

Using the formula \( A(r) = \pi r^2 \), substitute \( r = 8 \). This gives us the expression \( A(8) = \pi \times 8^2 \).

Step 4: Calculate \(8^2\)

Calculate the square of the radius: \(8^2 = 64\).

Step 5: Express the Area with Pi

Substitute the squared radius back into the formula: \( A(8) = \pi \times 64 \). The area of the circular garden is \( 64\pi \) square feet.

Key Concepts

Radius of a Circle

The radius of a circle is a fundamental element of circle geometry. It is defined as the distance from the center of the circle to any point along its edge or circumference. Understanding the radius is crucial because it is often the starting point for various calculations regarding a circle, such as its area or circumference.

• The radius is half of the diameter, which is the longest straight line that can be drawn across a circle, passing through the center.
• Knowing the radius allows you to use important geometric formulas, like those for finding the area or circumference of a circle, with ease.

In our exercise, the radius is provided as 8 feet. By knowing this, we can easily calculate the circle's area using the appropriate formula.

Circle Geometry

Circle geometry involves various interesting principles and measurements unique to circles. Let's explore a few essential concepts:

• The Circumference: This is the complete outer edge or boundary of a circle. The formula for circumference, when the radius is known, is given by \( C = 2\pi r \).
• The Area: This considers the entire space inside the circle. An important formula is \( A = \pi r^2 \), where \( r \) is the radius.
• Diameter: This is twice the length of the radius. Hence, \( D = 2r \).
• Central Angle: An angle whose vertex is at the center of the circle, separating the circle into two arcs or sections.

In this problem, we focus on using the radius to find the area. Understanding how these elements interrelate is essential for solving problems related to circle geometry.

Geometry Formulas

Geometry is packed with formulas that help solve problems related to various shapes, including circles. When dealing with circle problems, here are some key formulas:

• Area of a Circle: The area is calculated using \( A(r) = \pi r^2 \). This formula tells us the size of the space inside the circle based on its radius.
• Circumference of a Circle: The circumference is \( C = 2\pi r \). This formula is used to determine the distance around the circle.
• Diameter of a Circle: The diameter is simply twice the radius, making it a straightforward calculation: \( D = 2r \).

These formulas allow us to transition seamlessly from knowing one measurement, like the radius, to finding other properties of the circle. In our exercise, the formula for the area of the circle \( A(r) = \pi r^2 \) was used to find the area, demonstrating how understanding and applying these formulas is key in circle geometry.