Problem 54 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The illumination from a light source varies inversely as the square of the distance from the light source. If you raise a lamp from 15 inches to 30 inches over your desk, what happens to the illumination?

Step-by-Step Solution

Answer

When the lamp is raised from 15 inches to 30 inches, the illumination decreases to a quarter of its original value due to the inverse square law.

Step 1: Understanding Inverse Square Law

An inverse square law signifies a physical quantity or strength decreases with the square of the distance. So, if the distance gets doubled, the physical quantity will get reduced to a quarter of its original value. In this case, illumination intensity is the physical quantity.

Step 2: Applying Inverse Square Law

Considering inverse square law, if the distance of the lamp from the desk is doubled (i.e., from 15 inches to 30 inches), then the illumination or intensity of light would get reduced to \((1/2)^2 = 1/4\) of its original value. This is because the illumination follows the inverse square law.

Step 3: Final Conclusion

Therefore, by raising the lamp from 15 inches to 30 inches, the illumination at the desk decreases to a quarter of the original illumination.

Key Concepts

Illumination

Illumination refers to the amount of light that falls on a surface. This is essential in contexts where adequate lighting is crucial, such as in offices or homes. Understanding illumination helps us adjust lighting to meet various requirements. The illumination of a surface is directly influenced by the light source's power and distance.
The relationship between illumination and distance is explained by the inverse square law. This law shows that as distance from the light source increases, the intensity of light decreases significantly. This is important because even slight adjustments in distance can lead to noticeable changes in light levels.
In everyday life, this concept is utilized by adjusting the height or distance of lamps and light sources to achieve desired illumination levels. Efficient lighting setups can enhance productivity, mood, and the overall ambiance of a space.

Distance and Intensity

When referring to light, 'intensity' is the brightness or strength of the light emitted from a source. In understanding the relationship between distance and light intensity, the inverse square law provides valuable guidance.
The law states that if you move twice as far away from a light source, the brightness or intensity becomes one-fourth. This is because light spreads over a larger area as you move away. The practical implication is that if you are setting up a workspace or any environment where light is crucial, the distance of the light source significantly affects how well-lit the area is.

• If you double the distance, intensity goes down to a quarter.
• If you halve the distance, intensity goes up four times.

These distances and their effects are easily illustrated if you think about standing close to a lamp versus standing further away. Close up, the light is bright and intense; far away, it is dimmer.

Physical Quantity Variation

Physical quantity variation describes how different physical quantities change or remain consistent under varying conditions. For illumination, this involves understanding not just light intensity but how it reacts to changes in distance based on the inverse square law.
For instance, if a lamp is pulled from 15 inches to 30 inches away from a surface, the illumination does not merely decrease by half but is reduced to a quarter. This variation is crucial in planning spaces that require specific lighting needs, ensuring areas are not overly dim or too bright.
In general, when dealing with physical quantities, it's vital to comprehend how they alter with changes in environmental factors. This knowledge helps in creating stable and accurate models in scientific experiments and everyday applications. The inverse square law is a classic example of how mathematical relationships simplify understanding these changes.