Problem 46 - Elementary Algebra

Question

Applications involving variation. The area of circle varies directly as the square of its radius. If the area of a circle with radius 7 centimeters is determined to be \(49 \pi\) square centimeters, then find the constant of proportionality.

Step-by-Step Solution

Answer

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

Step 1: Understanding the Relationship

The problem states that the area of a circle varies directly as the square of its radius. Mathematically, this is expressed as: \[ A = k imes r^2 \] where \( A \) is the area of the circle, \( r \) is the radius, and \( k \) is the constant of proportionality.

Step 2: Substitute Known Values

We're given that the area \( A = 49\pi \) square centimeters and radius \( r = 7 \) centimeters. Substituting these into the variation equation gives: \[ 49\pi = k imes 7^2 \] which simplifies to \[ 49\pi = k imes 49 \]

Step 3: Solve for the Constant of Proportionality

To find the constant \( k \), divide both sides of the equation by 49:\[ k = \frac{49\pi}{49} \] Simplifying this gives the constant of proportionality:\[ k = \pi \]

Key Concepts

Understanding Direct Variation

When we talk about direct variation, it's like describing a relationship where one quantity changes directly with another. In direct variation, when one quantity increases, the other does too, and when one decreases, so does the other. It's like they are synchronized! Direct variation is mathematically expressed as: \[ y = kx \] Here, \( y \) varies directly with \( x \), and \( k \) is the constant of proportionality. A real-world example of direct variation could be the relationship between the distance traveled by a car and the time spent traveling at a constant speed. If the speed (addressed as the constant of proportionality) remains unchanged, doubling the time doubles the distance. In our original exercise, direct variation is used to describe how the area of a circle changes with the square of its radius. If the radius increases, the area increases as a square function of the radius.

Decoding the Constant of Proportionality

The constant of proportionality is a key factor in the equation of direct variation. It determines the rate at which one variable changes in relation to another. The equation \( y = kx \) shows us that \( k \) dictates how much \( y \) will be when \( x \) is a certain value. In many direct variation scenarios, finding the constant \( k \) helps bring clarity to the relationship. In our exercise, we found \( k \) between the area of a circle and the square of its radius, resulting in \( k = \pi \).

• Constant \( k \) gives the relationship its specific scale.
• Without \( k \), the equation wouldn’t hold true for the given values.

The beauty lies in its consistency - it reflects how tightly the variables are linked in a uniform manner throughout the equation. Understanding \( k \) allows us to predict one variable based on the other, making it indispensable for solving real-world problems.

Exploring the Area of a Circle

The area of a circle is a classic example in mathematics, showing a geometric application of direct variation. It relates directly to the square of its radius, forming an equation that is essential for various calculations. For any circle, its area \( A \) is given by: \[ A = \pi r^2 \] Here, \( \pi \) (approximately 3.1416) serves as the constant of proportionality, expressing the fixed relationship between the variables.

• The area increases quadratically as the radius increases.
• If you double the radius, the area multiplies by a factor of four, since \((2r)^2 = 4r^2\).

Understanding this formula doesn't just help in academic settings. It has practical implications in fields ranging from architecture to engineering, where precise calculations of areas are critical. The inception of this formula originates in the fundamental properties of circles, allowing for elegant mathematical descriptions of their spaces.