Problem 45 - Elementary Algebra

Question

Applications involving variation. The circumference of a circle is directly proportional to its radius. If the circumference of a circle with radius 7 centimeters is measured as \(14 \pi\) centimeters. then find the constant of proportionalitv.

Step-by-Step Solution

Answer

The constant of proportionality is \( 2\pi \).

Step 1: Understanding Direct Proportionality

In direct proportionality, if one variable increases, the other variable increases proportionally. For our case, the relationship between the circumference (C) and the radius (r) of the circle can be expressed as: \( C = k \cdot r \), where \( k \) is the constant of proportionality.

Step 2: Identify Given Values

We know the circumference \( C = 14 \pi \) and the radius \( r = 7 \) centimeters. These values will help us find the constant \( k \).

Step 3: Set Up the Formula

Substitute the given values into the proportionality equation: \( 14\pi = k \cdot 7 \). This equation will allow us to solve for \( k \).

Step 4: Solve for Constant

To isolate \( k \), divide both sides by 7: \( k = \frac{14\pi}{7} \). Simplify the expression: \( k = 2\pi \).

Step 5: Verify the Solution

Confirm the computation is accurate: by substituting \( k = 2\pi \) and \( r = 7 \) back into the original equation, \( C = k \cdot r = 2\pi \times 7 = 14\pi \), which matches the given circumference.

Key Concepts

Circumference of a Circle

The circumference of a circle is the distance around the circle's edge. Think of it as the circle's perimeter. It's important because it measures how long it takes to walk around the circle.
The formula for the circumference (\( C \)) relates to the radius (\( r \)) through the equation \( C = 2\pi r \).
In this equation, \( \pi \) (pi) is a mathematical constant approximately equal to 3.14159. It remains the same no matter the circle's size, making it a crucial number in geometry. By using the formula, we can quickly find out the length of a string if we were to wrap it neatly around the circle.
If you know the radius, you just multiply it by 2 and \(\pi\) to get the circumference. Understanding this relationship is key to solving many mathematical problems involving circles.

Constant of Proportionality

The constant of proportionality is a fixed value that describes how one quantity changes in relation to another. When we talk about things being directly proportional, we mean that our main value, such as circumference, increases at the same rate as the radius.
In mathematical terms, if \( C = k \cdot r \), then \( k \) is the constant of proportionality.
In our exercise, since the circumference \( C \) and the radius \( r \) are directly proportional, \( k = \frac{C}{r} \) represents how many units of circumference we get per unit increase in radius.
To find \( k \), we insert the values of the given circumference and radius into the equation. So, if the circumference is \( 14\pi \) and the radius is 7, solving \( k = \frac{14\pi}{7} \) confirms \( k = 2\pi \). This shows that for every one unit increase in radius, the circumference increases by \( 2\pi \) units.

Radius of a Circle

The radius of a circle is half the distance across the circle, taken through the center point. It's a fundamental measurement because it helps us find other important properties of the circle.
If a circle is a pizza, the radius is a line from the crust straight to the middle of the pizza.
Knowing the radius allows us to calculate both the circumference and the area of the circle. With the formula \( C = 2\pi r \), if you know the radius, you can see how it directly affects the circumference.
In our exercise, the radius was given as 7 centimeters. From this, the circumference was calculated directly. Understanding the role of the radius helps demystify how circles work and solidifies your mathematical foundation.