Problem 78 - Algebra and Trigonometry

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) Intensity at 30 ft is 7.866 lumens. (b) Depth for intensity 5 lumens is 86.64 ft.

Step 1: Identify the Formula

The given formula to calculate the light intensity at any depth is \(I = 10e^{-0.008x}\). We will use this equation to solve both parts of the exercise.

Step 2: Solve for Intensity at 30 Feet

Substitute \(x = 30\) into the formula: \(I = 10e^{-0.008 \times 30}\). Calculating the exponent, we have \(-0.008 \times 30 = -0.24\). Thus, \(I = 10e^{-0.24}\). Use a calculator to find \(e^{-0.24} \approx 0.7866\). So, \(I \approx 10 \times 0.7866 = 7.866\) lumens.

Step 3: Set Equation for Intensity of 5

We need to find \(x\) such that the intensity \(I = 5\). Set up the equation \(5 = 10e^{-0.008x}\).

Step 4: Solve for Depth \(x\)

First, divide both sides by 10 to isolate the exponential term: \(0.5 = e^{-0.008x}\). Take the natural logarithm of both sides: \(\ln(0.5) = -0.008x\). Compute \(\ln(0.5) \approx -0.6931\). Now solve for \(x\): \(-0.6931 = -0.008x\). Divide by \(-0.008\) to obtain \(x \approx 86.64\) feet.

Key Concepts

Light Intensity

In the context of lakes, light intensity is a measure of how much light penetrates through the water at different depths. This is a crucial factor because it affects the growth of aquatic plants, which need light for photosynthesis. The primary formula used to describe light intensity at a given depth in water is \( I = 10 \times e^{-0.008x} \),where \( I \) represents light intensity measured in lumens, and \( x \) denotes the depth in feet.

• The initial constant (10 lumens) represents the surface light intensity.
• The exponential decay factor \( e^{-0.008x} \) shows how light diminishes as depth increases.

Light decreases exponentially with depth, meaning it drops dramatically as it goes deeper, which can influence various underwater processes.

Depth Measurement

Understanding how light intensity changes with depth is essential in aquatic studies and is typically measured in feet or meters. Calculating depth in terms of light intensity involves solving equations derived from exponential decay patterns. For example, to find the depth where the light intensity is a specific value, such as 5 lumens, we rearrange the formula:\[I = 10 \times e^{-0.008x}\]When solving for \(x\), we take natural logarithms on both sides, using the formula's inverse to bring out the combination of constants and variables. As shown in the solution, once you set the intensity to 5 lumens and solve for \(x\), you find that light at this intensity reaches a depth of about 86.64 feet.

Environmental Science

In environmental science, understanding light intensity at different depths is key to studying aquatic ecosystems. Varied light intensity affects:

• The growth and health of aquatic plants.
• The development and movement of algae and other microorganisms.

Factors such as water clarity and quality play a role in light distribution and can significantly impact biodiversity. By analyzing light penetration, scientists assess the health of water bodies and implement conservation strategies.

Biodiversity

Biodiversity in aquatic environments is highly dependent on light availability, as light is needed for photosynthesis, which is a primary energy source for many organisms. Ensuring sufficient light at various water depths is crucial to maintain a healthy and diverse range of species, especially submerged plants called macrophytes.
Macrophytes serve as facilitators for fish and other aquatic life, providing habitat and contributing to the overall health of an ecosystem. When the water transparency is inadequate, it can lead to decreased plant growth, affecting the entire food chain due to diminished oxygen levels and habitat quality.