Problem 44 - 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 \(d .\) If two people are viewing the same light source, and one person is three times the distance from the light source as the other person, compare the light intensities that the two people observe.

Step-by-Step Solution

Answer

The first person sees the light 9 times brighter than the second person.

Step 1: Understand the Problem

We are given that the intensity of light, \( I \), is inversely proportional to the square of the distance from the light source, \( d \). We need to compare the intensities observed by two people at different distances from the same light source.

Step 2: Write the Proportional Relationship

The intensity \( I \) can be expressed as \( I = \frac{k}{d^2} \), where \( k \) is the proportionality constant. Here, \( d \) is the distance from the light source.

Step 3: Use Given Distances

Let the distance of the first person from the light source be \( d_1 = d \) and the distance of the second person be \( d_2 = 3d \) because they are three times further away than the first person.

Step 4: Calculate the Intensity for Each Person

For the first person, the intensity is \( I_1 = \frac{k}{d^2} \). For the second person, the intensity is \( I_2 = \frac{k}{(3d)^2} = \frac{k}{9d^2} \).

Step 5: Compare the Two Intensities

To compare the intensities, find the ratio \( \frac{I_1}{I_2} = \frac{\frac{k}{d^2}}{\frac{k}{9d^2}} = \frac{k}{d^2} \times \frac{9d^2}{k} = 9 \). Thus, \( I_1 = 9I_2 \).

Key Concepts

Intensity of Light

The concept of light intensity is a crucial aspect in understanding how light from sources like stars reaches us. Light intensity, often denoted as \(I\), is the measure of the power carried by light waves over a unit area. Imagine a lamp shining onto a surface; the intensity would describe how much light power hits each square unit of that surface. An important characteristic of intensity is that, when the source is a point source, it is inversely proportional to the square of the distance \(d\). This means as you move away from the light, the intensity decreases and vice versa.
To mathematically express this relationship, we use the formula: \(I = \frac{k}{d^2}\), where \(k\) is a constant that depends on the properties of the source. This inverse square law tells us that if the distance from the source doubles, the intensity becomes one-fourth of its original value. Understanding light intensity is important in astronomy, photography, and various engineering fields where light measurement is essential.

Distance and Brightness

Distance and brightness have an intertwined relationship due to the inverse square law. This law explains why a star may appear less bright even if it is quite luminous—simply because it is far away. Brightness, the apparent intensity of a light source, decreases dramatically with distance.
Let's consider how we observe objects on Earth: a light bulb appears dimmer the further we move away from it. This same principle applies to celestial bodies. If you double your distance from a light source, you aren't receiving twice less light, but significantly less—only a quarter of the light reaches you compared to your original distance. This decrease in brightness relates directly to the square of the distance.
In our original exercise, two people view the same star. The person three times farther away has to contend with a much dimmer star, seeing only one-ninth of the brightness because \((3)^2 = 9\). This relationship is how astronomers can deduce distances based on observed brightness.

Compare Light Intensities

Comparing light intensities can be a straightforward task if we understand the inverse proportionality of intensity with distance squared. In practical terms, this means we can determine how much brighter one location is compared to another simply by knowing their distances to the light source.
Consider the given scenario: person A is at distance \(d\), while person B is at distance \(3d\). Using the formula \(I = \frac{k}{d^2}\), we can calculate their respective light intensities. Person A receives intensity \(I_1 = \frac{k}{d^2}\), whereas person B sees intensity \(I_2 = \frac{k}{(3d)^2} = \frac{k}{9d^2}\).
This implies that \(I_1\) is nine times \(I_2\), as shown by the ratio \(\frac{I_1}{I_2} = 9\). Hence, the light appears nine times brighter to person A than to person B. Such comparisons are powerful tools not just in telescopic observations but in any field analyzing the spread or focus of light, such as lighting design and safety effectiveness in public spaces.