Problem 66 - A Graphical Approach to College Algebra

Question

The age in years of a female blue whale is approximated by $$=-2.57 \ln \left(\frac{87-L}{63}\right)$$ where \(L\) is its length in feet. (a) How old is a female blue whale that measures 80 feet? (b) Estimate the length of a female blue whale that is 4 years old. (c) The equation that defines t has domain 24

Step-by-Step Solution

Answer

(a) 5.6 years, (b) 73.7 feet, (c) Domain: 24 < L < 87.

Step 1: Calculate the Age for a Given Length

To find the age of a female blue whale measuring 80 feet, substitute \(L = 80\) into the expression for the age. The equation becomes \[t = -2.57 \ln \left(\frac{87-80}{63}\right)\]First, calculate the inside of the logarithm: \(87 - 80 = 7\). Then, divide by 63: \(\frac{7}{63} = \frac{1}{9}\). Now, evaluate the natural logarithm: \(\ln \left(\frac{1}{9}\right) \approx -2.197\). Multiply by -2.57 to find the age: \[t = -2.57 \times (-2.197) \approx 5.64\]So, the whale is approximately 5.6 years old.

Step 2: Estimate Length for a Given Age

To estimate the length for a 4-year-old whale, start with the equation \[4 = -2.57 \ln \left(\frac{87-L}{63}\right)\]Divide both sides by -2.57: \[\ln \left(\frac{87-L}{63}\right) = \frac{4}{-2.57} \approx -1.556\]Exponentiate both sides to isolate the fraction: \[\frac{87-L}{63} = e^{-1.556} \approx 0.211\]Now solve for \(L\): 1. Multiply both sides by 63: \(87-L \approx 13.293\)2. Subtract 87 from both sides: \(-L \approx 13.293 - 87 = -73.707\)3. Divide by -1 to find \(L\): \(L \approx 73.707\).Therefore, the estimated length is approximately 73.7 feet.

Step 3: Determine the Domain of the Equation

The domain of the equation is determined by the expression inside the logarithm. The argument \(\frac{87-L}{63}\) must be greater than 0, so we have \(87-L > 0\). Solving for \(L\) gives \(L < 87\).Additionally, the argument cannot equal 0, so it must also be positive. Therefore \(L > 87 -63 = 24\).Thus, the domain is \(24 < L < 87\), ensuring the logarithm is defined.

Key Concepts

Understanding Logarithmic Functions

Logarithmic functions help us understand relationships where one variable depends on another in a multiplicative way. A logarithm, expressed as \(\ln(x)\), is the power to which a base number (usually \(e\) in natural logs) must be raised to produce the number \(x\).
This concept is essential in the equation given in the problem, where the age of the blue whale is determined through a logarithmic function. Here, you use the logarithm to simplify complex relationships.
Why use logarithms? They are extremely useful for solving equations where the variable is in the exponent. Logarithms transform multiplicative processes into linear ones, making them easier to work with.

• This means finding the age of a blue whale involves determining how its length affects a logarithmic function as it scales over time.
• The logarithm helps us to find out the scaling nature of real-world phenomena like growth or decay.

Understanding this relationship ensures you grasp how logarithmic calculations reflect real-world biological patterns, especially in fields like biology and environmental science.

Exploring Domain and Range in Logarithmic Contexts

When working with logarithmic functions, knowing the domain and range is crucial for ensuring the function is correctly defined. The domain refers to all possible input values \(L\) (the length in this context), while the range refers to all possible output values \(t\) (the whale's age).
The domain of a logarithmic function like \(\ln\left(\frac{87-L}{63}\right)\) needs the argument inside the log to be greater than zero. This means that \(\frac{87-L}{63} > 0\) or simply, \(87 - L > 0\). Thus, \(L < 87\).
However, since \(L\) correlates to physical length, there's a lower limit as well. We calculate this to be \(L > 24\), ensuring the whole function remains logical and relevant to real world conditions.

• Range is determined by calculating possible ages, ensuring they align with what we understand about whale growth.
• This allows us to understand limitations and possibilities of the function's outputs, ensuring predictions (like age from length) remain realistic.

Understanding the domain ensures that you only consider lengths that make sense biologically, while range calculations reflect viable age predictions.

Grasping Exponential Equations

Exponential equations, such as those involving \(e^x\), are key to solving growth and decay problems. These equations feature prominently when reversing a logarithmic function, as seen when estimating the whale's length from its age.
An exponential equation allows us to shift from logarithmic expressions back into user-friendly numeric outputs. For example, once the logarithm \(\ln\left(\frac{87-L}{63}\right)\) is isolated, you exponentiate to solve for \(\frac{87-L}{63}\). This step reveals the real-world component you need - the whale's physical length.
Why exponentials matter: They provide insight into how continuous growth processes operate, especially useful in biology for measuring natural phenomena.

• The declination to exponential format allows the interpretation of a function's growth pattern over time.
• This is vital in understanding how certain biological attributes increase or decrease exponentially rather than linearly.

Grasping these concepts ensures you reliably decode the functional relationships at play, vital for accurate biological assessments and predictions.