Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

We found in Section 1.3 that light intensity, \(I_{d},\) as a function of depth, \(d\) was given by $$I_{d}=0.4 \times 0.82^{d}$$ Find \(k\) so that \(I_{d}=0.4 e^{k \times d}\).

Step-by-Step Solution

Answer

The value of \(k\) is approximately -0.19867.

Step 1: Recognize the equation forms

The given expression for light intensity is a function of depth, represented as both an exponential decay in base 0.82, and an exponential function with base e. Our goal is to convert the equation from base 0.82 to base e.

Step 2: Set the equations equal to each other

We want the equation in terms of the exponential with base e, so we set the two forms equal to each other: \(0.4 \times 0.82^d = 0.4 \times e^{k \cdot d}\). Because the coefficients 0.4 are the same on both sides, they cancel out, simplifying our equation to \(0.82^d = e^{k \cdot d}\).

Step 3: Equate the exponents and simplify

Since the bases are different (0.82 and e), set the exponents equal: \(0.82^d = e^{k \cdot d}\). This leads to \(d \cdot \ln(0.82) = d \cdot k\). Simplifying, it leaves us \(k = \ln(0.82)\).

Step 4: Compute the value of k

Now calculate \(k\) using the natural logarithm: \(k = \ln(0.82) \approx -0.19867\). You can use a scientific calculator to find that the natural logarithm of 0.82 is approximately -0.19867.

Key Concepts

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. It's commonly observed in processes like radioactive decay, cooling temperatures, and, as in our exercise, the attenuation of light as it penetrates deeper into a medium.
To model exponential decay, we frequently use equations of the form \(y = a \times b^x\), where \(a\) represents the initial quantity, \(b\) dictates the decay factor, and \(x\) is the variable of interest (in this case, depth \(d\)). Here, for light intensity, \(a = 0.4\) and \(b = 0.82\).
When \(b\) is between 0 and 1, this indicates a decrease. Hence, 0.82 means that the light intensity reduces with each unit increase in depth. As depth increases, the light weakens, illustrating the exponential decay phenomenon in our scenario.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a logarithm with a special base \(e\), where \(e \approx 2.71828\). It's an especially useful mathematical tool in various scientific fields due to its unique properties and relationship with exponential functions.
In our exercise, we convert an exponential decay equation with base 0.82 to an equivalent exponential equation with base \(e\). This is achieved by using the property that \(a^x = e^{x \cdot \ln(a)}\).

• This conversion is crucial when solving equations that involve different bases.
• Recognizing \(\ln\) allows for more streamlined calculations and solutions.
• The property helps compare models or equations, as was necessary in converting \(0.82^d\) to \(e^{k \cdot d}\).

Calculating \(\ln(0.82)\) provides the decay rate \(k\) for our equation, compactly integrating the decay factor into a form compatible with base \(e\).

Light Intensity Modeling

Light intensity modeling is a practical application of exponential functions, used to describe how light diminishes as it travels through a medium like water or air. Scientists and engineers use models like these to predict light behavior and make calculations for various environmental and technological applications.
For example, consider our initial light intensity formula, \(I_d = 0.4 \times 0.82^d\), which effectively models how light becomes dimmer with each layer it passes through. The equation translates to a more general form using base \(e\): \(I_d = 0.4 \times e^{k \times d}\).

• This allows us to utilize calculus and other analytical techniques more easily when the natural exponential base is involved.
• Such modeling is crucial in fields like marine biology, where understanding light penetration in water is vital for studying underwater ecosystems.
• Additionally, it can impact climate studies, photographic exposure, and energy conservation in building designs.

Hence, converting different exponential bases ensures that mathematical tools and insights can be universally applied across various applications.