Problem 29 - Elementary Algebra

Question

Suppose that the length of a radius of a circle is the same as the length of a side of a square. If the circumference of the circle is \(15.96\) centimeters longer than the perimeter of the square, find the length of a radius of the circle. (Use \(3.14\) as an approximation for \(\pi\).)

Step-by-Step Solution

Answer

The radius of the circle is approximately 7 cm.

Step 1: Understanding the Problem

We have a circle and a square with the radius of the circle equal to the side length of the square. The circumference of the circle is known to be 15.96 cm more than the perimeter of the square. Our task is to find the length of the radius.

Step 2: Setting Up Equations

Let the radius of the circle be \( r \) cm. Then, the circumference of the circle is \( 2\pi r \) and the side of the square is also \( r \). The perimeter of the square is \( 4r \). We are given that \( 2\pi r = 4r + 15.96 \) cm.

Step 3: Substitution of \( \pi \)

Substitute \( \pi = 3.14 \) into the equation: \[ 2(3.14)r = 4r + 15.96 \] Simplify to get: \[ 6.28r = 4r + 15.96 \]

Step 4: Solving for \( r \)

To solve for \( r \), reorganize the equation:\[ 6.28r - 4r = 15.96 \]Simplify to get:\[ 2.28r = 15.96 \] Divide both sides by \( 2.28 \):\[ r = \frac{15.96}{2.28} \]

Step 5: Final Calculation

Calculate \( r \):\[ r \approx \frac{15.96}{2.28} \approx 7 \] Thus, the radius of the circle is approximately 7 cm.

Key Concepts

Circle and Square Geometry

In this exercise, we have a circle and a square sharing a unique relationship. The radius of the circle matches the side length of the square. This setup is quite common in geometry, where one dimension serves dual purposes in different shapes.

The radius of the circle is a straight line from the center to its perimeter. This measurement is crucial as it helps to compute various properties of the circle, such as its circumference and area. Conversely, the side of the square is one of the four equal lengths enclosing the square's area.

The relationship between these two shapes in this problem creates an opportunity to apply geometric formulas to solve for unknown dimensions using their interconnected properties.

Equation Setup

Setting up equations is a fundamental skill in algebra problem-solving. Here, we establish a connection between the circle and square using the given information. Denote the radius of the circle as \( r \). Hence, the side length of the square is also \( r \).

The formula for the circumference of the circle is \( 2\pi r \), while the square's perimeter is \( 4r \). The problem states that the circle's circumference exceeds the square's perimeter by 15.96 cm. We create the equation:

• \( 2\pi r = 4r + 15.96 \)

This equation ties together the geometry of both shapes and sets the stage for solving for \( r \).

Problem Understanding

Understanding the given problem is often the first and most crucial step in finding a solution. Here, we carefully consider how the circle and square relate to one another through their shared dimension - the radius and side length.

The problem provides a unique twist: the circle's circumference is stated as being 15.96 cm more than the square's perimeter. Such information requires us to interpret the stated relationships carefully and reflect them accurately in our equation. Missing this step can lead to misinterpretations and further difficulties in solving the exercise. Remember, always lay out what you know first and clarify every mathematical term before rushing into calculations.

Circumference and Perimeter

The terms "circumference" and "perimeter" may sound similar, but they apply to different shapes. The circumference is the total distance around a circle, calculated with the formula \( 2\pi r \), where \( r \) is the radius.

Meanwhile, the perimeter is the sum of all side lengths of a polygon, such as a square. For a square, this is \( 4r \), since it has four equal sides.

Understanding the difference and correctly applying these concepts allows us to use the provided data effectively. By substituting the given \( \pi = 3.14 \) into our initial equation, we simplified the problem to solve for the radius effectively. Getting confidence in these foundational definitions ensures a smooth path in applying more complex problem-solving strategies.