Problem 17 - Elementary Algebra

Question

The relationship of the radius of a circle, \(x\), and the circumference of the circle, \(y\), is a direct variation. The radius of a circle is \(10 \mathrm{~cm}\), and the circumference is \(62.8 \mathrm{~cm}\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the circumference of a circle with a radius of \(20 \mathrm{~cm}\).

Step-by-Step Solution

Answer

a. \(k = 6.28\); b. \(y = 6.28x\); c. \(125.6 \text{cm}\).

Step 1: Understand Direct Variation

In a direct variation, the relationship between two variables can be expressed as \(y = kx\), where \(k\) is the constant of proportionality.

Step 2: Identify Given Values

Given the radius \(x = 10 \text{cm}\) and the circumference \(y = 62.8 \text{cm}\), we need to find the value of \(k\).

Step 3: Find the Constant of Proportionality

Use the direct variation formula \(y = kx\). Substitute the given values to find \(k\): \ 62.8 = k \times 10\. Solving for \(k\), we get \(k = \frac{62.8}{10} = 6.28\).

Step 4: Write the Equation

Now that we have \(k = 6.28\), the equation representing this relationship is \(y = 6.28x\).

Step 5: Find the Circumference for a New Radius

To find the circumference when the radius is \(20 \text{cm}\), substitute \(x = 20\) into the equation \(y = 6.28x\). This gives \(y = 6.28 \times 20 = 125.6 \text{cm}\).

Key Concepts

radius and circumference

The radius and circumference of a circle have a special relationship known as direct variation. This means they are proportional to each other. The radius is the distance from the center of the circle to any point on its edge, while the circumference is the total distance around the circle. Whenever you know the radius, you can find the circumference by multiplying the radius by a constant. This constant is found using the formula for the circumference of a circle: \( C = 2\pi r \). To simplify this, when we know the radius, the circumference can be directly calculated.

constant of proportionality

In direct variations, the constant of proportionality is key. It's represented by the symbol 'k'. In our circle problem, the radius and circumference vary directly. This means the circumference (y) is equal to the radius (x) multiplied by the constant (k). From the exercise, we know:
\[ y = kx \]
To find 'k', we use the given values: radius (10 cm) and circumference (62.8 cm). Plugging these into the equation:
\[ 62.8 = k \times 10 \]
Solving for 'k', we get: \[ k = \frac{62.8}{10} = 6.28 \]
This value, 6.28, remains the same for any circle. It shows how much the radius needs to be multiplied to get the circumference.

algebraic equation

An algebraic equation allows us to easily calculate unknown values. From the above calculation, we found the constant of proportionality 'k' to be 6.28. Using this, we can write a general equation for the relationship between the radius and circumference of any circle:
\[ y = 6.28x \]
This equation tells us that if we know the radius of a circle, we can find the circumference by multiplying the radius by 6.28. For example, if the radius is 20 cm, we can calculate the circumference as follows:
\[ y = 6.28 \times 20 = 125.6 \]
Therefore, the circumference will be 125.6 cm. This simple equation helps quickly find solutions and understand the relationship between the radius and circumference.