Problem 42 - Algebra 2

Question

Astronomers can use the brightness of two light sources, such as stars, to compare the distances from the light sources. The intensity, or brightness, of light I is inversely proportional to the square of the distance from the light source \(d .\) Write an equation that represents this situation.

Step-by-Step Solution

Answer

\( I = \frac{k}{d^2} \) represents the brightness-distance relationship.

Step 1: Understanding the Relationship

The problem states that the intensity (brightness) of light \(I\) is inversely proportional to the square of the distance \(d\) from the light source. This means that as the distance from the light source increases, the intensity decreases in relation to the square of that distance.

Step 2: Using the Inverse Proportionality Concept

Since intensity is inversely proportional to the square of the distance, we can express this relationship using the proportionality constant \(k\). The formula for inverse proportionality is: \[ I = \frac{k}{d^2} \] where \(k\) is a constant that depends on the specifics of the light source and intensity.

Step 3: Formulating the Equation

By incorporating the constant \(k\) into the relationship derived from the inverse proportionality, the equation representing the brightness and distance can be written as: \[ I = \frac{k}{d^2} \] This equation shows that the intensity \(I\) varies inversely as the square of the distance \(d\) from the light source.

Key Concepts

Intensity of Light

The intensity of light refers to how bright a light source appears from a given point. It can be thought of as the concentration of light energy over a surface area. For practical purposes, intensity is often measured in units like lumens or candela depending on the context. In astronomy, intensity is crucial because it helps astronomers determine distances to stars and other celestial objects.

Factors such as the power of the light source and the distance from it affect the intensity. A key aspect here is that light spreads out as it travels, meaning that the farther you are from the source, the less intense it appears. This is due to the light energy being spread over a larger area as the distance increases. Understanding how the intensity diminishes with distance helps astronomers make precise calculations regarding celestial distances.

Square of the Distance

When discussing the square of the distance, we are referring to the mathematical relationship where the variable of distance is squared or raised to the power of two. This means if the distance from a light source is represented as \(d\), the square of that distance is \(d^2\).

In the context of light intensity, this squared relationship is central to understanding why light intensity decreases so rapidly with distance. As the distance from the light source doubles, the light's intensity becomes a quarter because \((2d)^2 = 4d^2)\).

This principle is applicable not only in astronomy but in various fields dealing with radiant energy such as physics and engineering.

Proportionality Constant

The proportionality constant \(k\) in the equation \(I = \frac{k}{d^2}\) is a crucial factor that determines specific characteristics of the light source. It acts like a bridge connecting intensity with distance, giving a measure that adjusts the formula for real-world applications.

This constant has units that depend on the units used for intensity and distance and provides value in real-world calculations. For instance, \(k\) could involve factors such as the power output of the light source and environmental factors like atmospheric conditions.

In scientific studies, precisely knowing \(k\) helps to accurately model how light from stars and other celestial bodies behaves over vast distances of space. Understanding \(k\) allows for nuanced analysis and accurate conclusions in observational astronomy.