Problem 79 - Algebra for College Students

Question

Set up an equation and solve each of the following problems. The sum of the areas of a circle and a square is \((16 \pi+\) 64) square yards. If a side of the square is twice the length of a radius of the circle, find the length of a side of the square.

Step-by-Step Solution

Answer

The side length of the square is 8 yards.

Step 1: Understand the problem

We are given the sum of the areas of a circle and a square, which is \(16\pi + 64\) square yards. We need to find the length of a side of the square if the side is twice the radius of the circle.

Step 2: Define variables

Let \( r \) be the radius of the circle. Then, the side length of the square, \( s \), is given by \( s = 2r \).

Step 3: Write the area formulas

The area of the circle is \( \pi r^2 \). The area of the square is \( s^2 = (2r)^2 = 4r^2 \).

Step 4: Set up the equation

The sum of the areas is equal to \( 16\pi + 64 \), so the equation is: \[ \pi r^2 + 4r^2 = 16\pi + 64 \]

Step 5: Combine like terms in the equation

Group the terms with \( r^2 \) together: \[ r^2(\pi + 4) = 16\pi + 64 \]

Step 6: Solve for \( r^2 \)

Divide both sides by \( \pi + 4 \):\[ r^2 = \frac{16\pi + 64}{\pi + 4} \]

Step 7: Substitute numerical values

Calculate the value: \[ r^2 = \frac{16\pi + 64}{\pi + 4} \ = 16 \]

Step 8: Find the radius \( r \)

Take the square root of both sides: \[ r = \sqrt{16} = 4 \]

Step 9: Calculate the side of the square

Since \( s = 2r \), substitute \( r = 4 \):\[ s = 2 \times 4 = 8 \]

Key Concepts

Understanding Equations in Problem Solving

Equations are fundamental in algebra and essential for problem solving. They help us model and find solutions to real-world scenarios.
An equation represents a relationship where two expressions are equal. In the exercise, we set up an equation using the given areas of the circle and the square.

• For the circle, the area is given by the formula: \( \pi r^2 \) where \( r\) is the radius.
• For the square, the area formula is \( s^2\) where \( s \) (the side of the square) equals \( 2r \).

Our equation to find the total area is: \( \pi r^2 + 4r^2 = 16\pi + 64 \). This allows us to analyze and solve for missing variables.
Understanding and manipulating equations is key to deducing unknowns in algebraic problems.

Exploring Geometry: Circle and Square

Geometry plays a significant role in this exercise. It involves understanding the spatial properties of shapes like circles and squares.
A key aspect is the formulas used to define the area of these geometrical figures. For example:

• The area of a circle is calculated as \( \pi r^2 \), representing all the space inside its circumference.
• The square's area, given by \( s^2 \), covers all the space within its four equal sides.

In this problem, by equating these areas' sum to \( 16\pi + 64\), we integrate geometry into algebra. This allows further analysis using properties and equations related to each shape.

Role of Variables in Algebra

Variables are symbols or letters that stand in for unknown quantities in equations. They are crucial for formulating expressions and solving algebraic problems.
In the given exercise:

• The variable \( r \) represents the radius of the circle.
• The equation \( s = 2r \) defines the relationship between the square's side and the circle's radius, using \( s \) as another variable.

These variables help us express the areas and their relationship. Once we derive the equation, solving for these variables allows us to find the real-world measurements needed, like the length of the square's side. Understanding and correctly using variables is essential in algebra for converting complex problems into manageable calculations.