Problem 84 - Precalculus Mathematics for Calculus

Question

Transparency of a Lake Environmental scientists measure the intensity of light at various depths in a lake to find the "transparency" of the water. Certain levels of transparency are required for the biodiversity of the submerged macrophyte population. In a certain lake the intensity of light at depth \(x\) is given by $$I=10 e^{-0.008 x}$$ where \(I\) is measured in lumens and \(x\) in feet. (a) Find the intensity \(I\) at a depth of \(30 \mathrm{ft}\). (b) At what depth has the light intensity dropped to \(I=5 ?\)

Step-by-Step Solution

Answer

(a) 7.87 lumens at 30ft; (b) Depth is approximately 86.64 feet.

Step 1: Calculate Intensity at 30ft

Given the formula for light intensity \( I = 10 e^{-0.008 x} \), we need to find the intensity at \( x = 30 \) feet. Substitute \( x = 30 \) into the equation:\[I = 10 e^{-0.008 \times 30}\]This simplifies to:\[I = 10 e^{-0.24}\]Use a calculator to find \( e^{-0.24} \), which is approximately 0.7866. Thus, the intensity is:\[I \approx 10 \times 0.7866 = 7.866\]So, the intensity at 30 ft is approximately 7.87 lumens.

Step 2: Determine Depth for Intensity of 5

We need to find the depth \( x \) such that the intensity \( I = 5 \). Start with the equation:\[5 = 10 e^{-0.008 x}\]To solve for \( x \), first isolate the exponential term:\[e^{-0.008 x} = \frac{5}{10} = 0.5\]Now, take the natural logarithm of both sides:\[-0.008x = \ln(0.5)\]Calculate \( \ln(0.5) \), which is approximately -0.6931. Thus, the equation becomes:\[-0.008 x = -0.6931\]Solve for \( x \) by dividing both sides by -0.008:\[x = \frac{-0.6931}{-0.008} \approx 86.64\]Therefore, the light intensity drops to 5 lumens at a depth of approximately 86.64 feet.

Key Concepts

Exponential Decay in Light

When light travels through water, it becomes less intense as it penetrates deeper into the water. This decrease in light intensity is known as exponential decay. The concept of exponential decay is crucial in understanding how light behaves under water.

In exponential decay, the rate of decrease is proportional to the current value. The higher the initial light intensity, the faster it decays over a given depth. Mathematically, this can be expressed with an equation similar to the one given in the exercise:

• Intensity equation: \( I = 10 e^{-0.008 x} \)
• Where \( I \) is the light intensity in lumens, and \( x \) is the depth in feet.

The base of the natural logarithm, \( e \), signifies the exponential nature of the decay. As the depth \( x \) increases, the exponent \( -0.008x \) becomes larger in magnitude but negative, leading to a reduction in the intensity \( I \). Understanding this concept helps scientists predict how much light reaches different depths, which is essential for determining water transparency and the environmental health of aquatic ecosystems.

Transparency Measurement Techniques

Transparency in lakes and water bodies is vital for ecological health assessment. Scientists use specific techniques to measure how transparent a water body is, which involves determining how far light can penetrate the water before it is significantly reduced.

Some commonly used techniques include:

• Secchi Disk: A circular disk is lowered into the water until it is no longer visible. The depth at which it disappears indicates water clarity.
• Light Sensors: Devices measure light intensity at various depths to provide data on how much light penetrates through the water.
• Colorimetric Methods: These involve comparing the water's color with standards or using color sensors, which reflect the concentration of particulates or dissolved substances affecting transparency.

Each technique provides valuable insights, but combining them offers a more comprehensive view of a lake's transparency. By employing these measurements, scientists assess the potential for sunlight to reach underwater vegetation like submerged macrophytes, which require a certain amount of light to thrive and support biodiversity.

Biodiversity Requirements in Lakes

Lakes are home to complex ecosystems, where biodiversity is a key indicator of their health. Biodiversity refers to the variety and variability of life in a particular habitat, and in lakes, it includes different species of plants, fish, algae, and microorganisms.

For submerged macrophytes, a type of aquatic plant crucial for ecosystem health, sufficient light is necessary for photosynthesis. Healthy macrophyte populations promote:

• Increased habitat for aquatic organisms, providing food and shelter.
• Improved water quality through the absorption of nutrients and stabilization of sediments.
• Oxygenation of water, which benefits fish and other aquatic life.

If water transparency falls below certain levels, these plants cannot maintain their populations, leading to reduced biodiversity. This impacts the overall balance in aquatic environments and can lead to a decline in species that rely on these plants. Maintaining the right transparency level is thus critical for preserving biodiversity and ensuring healthy lake ecosystems.