Problem 62 - Calculus and its applications

Question

A beam of light enters a medium such as water or smoky air with initial intensity \(I_{0}\). Its intensity is decreased depending on the thickness (or concentration) of the medium. The intensity I at a depth (or concentration) of \(x\) units is given by $$I=I_{0} e^{-\mu x}$$ The constant \(\mu\left({ }^{u} m u "\right),\) called the coefficient of absorption, varies with the medium. Use this law for Exercises 62 and \(63 .\) Concentrations of particulates in the air due to pollution reduce sunlight. In a smoggy area, \(\mu=0.01\) and \(x\) is the concentration of particulates measured in micrograms per cubic meter \(\left(\mathrm{mcg} / \mathrm{m}^{3}\right)\) What change is more significant-dropping pollution levels from \(100 \mathrm{mcg} / \mathrm{m}^{3}\) to \(90 \mathrm{mcg} / \mathrm{m}^{3}\) or dropping them from \(60 \mathrm{mcg} / \mathrm{m}^{3}\) to \(50 \mathrm{mcg} / \mathrm{m}^{3}\) ? Why?

Step-by-Step Solution

Answer

The change from 60 mcg/m³ to 50 mcg/m³ is more significant.

Step 1: Understanding the Intensity Formula

The intensity of light transmitted through a medium is described by the formula \( I = I_0 e^{-\mu x} \). Here, \( I_0 \) is the initial intensity, \( \mu \) is the absorption coefficient, and \( x \) is the concentration of particulates. In this problem, \( \mu = 0.01 \). We need to evaluate the changes in intensity for different changes in \( x \).

Step 2: Calculate Intensity at 100 mcg/m³ and 90 mcg/m³

Substitute \( x = 100 \) and then \( x = 90 \) into the formula. For \( x = 100 \), \( I_{100} = I_0 e^{-0.01 \times 100} = I_0 e^{-1} \). For \( x = 90 \), \( I_{90} = I_0 e^{-0.01 \times 90} = I_0 e^{-0.9} \).

Step 3: Calculate Intensity Change from 100 mcg/m³ to 90 mcg/m³

Find the change in intensity: \( \Delta I_1 = I_{90} - I_{100} = I_0 e^{-0.9} - I_0 e^{-1} = I_0 (e^{-0.9} - e^{-1}) \).

Step 4: Calculate Intensity at 60 mcg/m³ and 50 mcg/m³

Substitute \( x = 60 \) and then \( x = 50 \) into the formula. For \( x = 60 \), \( I_{60} = I_0 e^{-0.01 \times 60} = I_0 e^{-0.6} \). For \( x = 50 \), \( I_{50} = I_0 e^{-0.01 \times 50} = I_0 e^{-0.5} \).

Step 5: Calculate Intensity Change from 60 mcg/m³ to 50 mcg/m³

Find the change in intensity: \( \Delta I_2 = I_{50} - I_{60} = I_0 e^{-0.5} - I_0 e^{-0.6} = I_0 (e^{-0.5} - e^{-0.6}) \).

Step 6: Comparing the Changes in Intensity

To determine which reduction in pollution is more significant, compare \( \Delta I_1 \) and \( \Delta I_2 \). Since \( e^{-0.9} > e^{-1} \) and \( e^{-0.5} > e^{-0.6} \), calculate numerically to compare these differences. Upon computation, \( e^{-0.5} - e^{-0.6} \) results in a larger value, implying a greater change in intensity for 60 mcg/m³ to 50 mcg/m³.

Key Concepts

Coefficient of Absorption

In understanding how light diminishes in intensity as it travels through different media, the coefficient of absorption, represented by \( \mu \), plays a critical role. It is a constant that quantifies the rate at which the medium "absorbs" the light, causing it to lose intensity. This coefficient is unique to each medium: it’s a small value in clear air because light can pass through with minimal obstruction, whereas it’s higher in denser media like cloudy water or smoke-filled air.
This concept shows up in the formula \( I = I_0 e^{-\mu x} \). Here, \( \mu \) identifies how potent the medium is at blocking or absorbing the light passing through. For light traveling through a smoky area, like in the problem described, a \( \mu \) value of \( 0.01 \) suggests moderate absorption, indicating that as the concentration \( x \) rises, the intensity \( I \) of the light diminishes at a measurable rate.

• Low \( \mu \) values suggest minimal absorption, meaning light travels farther without losing intensity.
• Higher \( \mu \) values indicate stronger absorption and greater intensity loss.

Recognizing how \( \mu \) impacts light intensity helps us predict and manage light behavior in various environments.

Intensity Change

Intensity change refers to the variation in light, sound, or any wave-like phenomena as they pass through a medium. In the given exercise, we are evaluating how light intensity alters when particulate concentration changes. The formula \( I = I_0 e^{-\mu x} \) is essential here to determine how much light gets through at different particulate concentrations.
To quantify this change, we need to plug values into the formula for different \( x \) concentrations. For this exercise:

• Initially, we calculate the light intensity for two concentrations: from \( 100 \text{ mcg/m}^3 \) to \( 90 \text{ mcg/m}^3 \).
• Then, assess it from \( 60 \text{ mcg/m}^3 \) to \( 50 \text{ mcg/m}^3 \).

By calculating the differences, we find that the drop from \( 60 \text{ mcg/m}^3 \) to \( 50 \text{ mcg/m}^3 \) results in a more significant change. Simply because the change in the exponent's value is larger in effect in this range. Thus, giving a better indication of how sensitive the intensity is to shifts in particulate matter within certain concentration ranges.

Pollution Levels Impact

Pollution levels have a direct impact on the environment by affecting how much sunlight can penetrate through our atmosphere. The presence and concentration of pollutants like particulates in the air reduce the intensity of sunlight arriving at Earth's surface. In the context of our formula, as we see, increasing \( x \) translates directly to a higher degree of absorption and a lower remaining light intensity, \( I \).
When we analyze changes in pollution levels, such as reducing from \( 100 \text{ mcg/m}^3 \) to \( 90 \text{ mcg/m}^3 \) versus \( 60 \text{ mcg/m}^3 \) to \( 50 \text{ mcg/m}^3 \), we uncover how different environmental conditions can affect solar penetration. This can significantly influence climate patterns, plant growth, and atmospheric conditions.
The greater reduction effect observed when going from \( 60 \text{ mcg/m}^3 \) to \( 50 \text{ mcg/m}^3 \) highlights the non-linear nature of exponential decay. It implies that even a small decrease in pollution at lower concentrations has a proportionally larger impact on improving light transmission.
Utilizing this understanding helps in optimizing environmental protection measures and strategizing pollution reduction to enhance solar exposure and ecological welfare.