Problem 25 - Algebra A Combined Function

Question

The intensity \(I\) of light varies inversely as the square of the distance \(d\) from the light source. If the distance from the light source is doubled (see the figure), determine what happens to the intensity of light at the new location.

Step-by-Step Solution

Answer

The intensity of light becomes one-fourth when the distance is doubled.

Step 1: Understanding the Inverse Relationship

The intensity of light, denoted as \(I\), varies inversely with the square of the distance \(d\) from the light source. This means the relationship can be expressed in the equation \( I = \frac{k}{d^2} \), where \(k\) is a constant that depends on the initial conditions.

Step 2: Setting Up the Initial Equation

Let's assume the initial distance from the light source is \(d\), and the corresponding intensity is \(I\). The equation is \( I = \frac{k}{d^2} \).

Step 3: Doubling the Distance

Now, we consider the distance is doubled. The new distance from the light source will be \(2d\). We want to find the new intensity \(I'\), so we plug \(2d\) into the equation: \(I' = \frac{k}{(2d)^2}\).

Step 4: Simplifying the New Intensity Equation

Simplify the expression for the new intensity: \(I' = \frac{k}{4d^2}\). This can be rewritten as \(I' = \frac{1}{4} \times \frac{k}{d^2}\), which simplifies further to \(I' = \frac{I}{4}\).

Step 5: Interpreting the Result

The new intensity \(I'\) is one-fourth of the original intensity \(I\). This shows that when the distance from the light source is doubled, the intensity of light at the new location decreases to a quarter of its original value.

Key Concepts

Intensity of Light

The intensity of light refers to the brightness or strength of the light at a given point. It's essentially a measure of how much light energy is hitting a unit area at a particular spot. The intensity depends on several factors, but one of the fundamental aspects is how far you are from the source of the light.
Light intensity is often symbolized as \(I\) in equations. It is crucial in various fields including photography, astronomy, and even biology, as it affects visibility and biological processes like photosynthesis.
In mathematical terms, when the light source is fixed and we talk about light traveling in all directions equally, its intensity naturally diminishes with increased distance. This decrease is governed by specific laws of physics discussed below.

Distance and Light Intensity

The connection between distance and light intensity might seem intuitive: move away from a light source, and it appears dimmer. This happens because as distance from the source increases, the spread of light covers a larger area. Thus, the same amount of light energy gets distributed over a more extensive territory.
We can depict this relationship mathematically by saying light intensity \(I\) is inversely proportional to the square of the distance \(d\) from the source. This can be expressed in the formula:

• \( I = \frac{k}{d^2} \)

where \(k\) is a constant.
The formula tells us that doubling the distance \(d\) results in a reduction of light intensity \(I\) by a factor of four, since \((2d)^2 = 4d^2\). Thus, increasing the distance causes the intensity decreases rapidly.

Inverse Square Law

The inverse square law is a scientific principle that describes how a given physical quantity behaves with respect to distance. It's one of the bedrock principles explaining how various types of energy or forces like light, sound, or gravity diminish with distance. When we say something "varies inversely with the square," it implies that as the distance increases, the observed effect (in this case, intensity) decreases proportionally to the square of the distance increase.
In the context of light, if you double the distance from a light source, the intensity becomes one-fourth. If you triple the distance, the intensity drops to just one-ninth. This is essential in designing lighting for spaces, understanding shadow formation, and even in calculating satellite data.
This law not only helps comprehend how light distributes but it also supports a broad range of real-world applications. From energy conservation in building design to estimating energy absorption in plants, recognizing how inverse square law impacts light intensity can lead to more efficient and practical solutions.