Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. The age in years of a female blue whale is approximated by $$t=-2.57 \ln \left(\frac{87-L}{63}\right).$$ 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) The whale is ~5.65 years old. (b) The length is ~73.71 feet. (c) Domain is set to ensure a positive argument for the logarithm and model consistency.

Step 1: Evaluating the Age for Given Length

To find the age of a female blue whale that measures 80 feet, substitute \(L = 80\) into the given formula: \[t = -2.57 \ln\left(\frac{87-80}{63}\right).\] First calculate the numerator: \(87 - 80 = 7\). Then compute the fraction: \( \frac{7}{63} = \frac{1}{9} \). Now find the natural logarithm: \( \ln\left(\frac{1}{9}\right) \approx -2.197. \) Finally, calculate \( t: t = -2.57(-2.197) \approx 5.65.\) Thus, the female blue whale is approximately 5.65 years old.

Step 2: Estimating Length for Given Age

To estimate the length of a whale that is 4 years old, we need to find \(L\) by rearranging the formula. Start by substituting \(t = 4\): \[4 = -2.57 \ln\left(\frac{87-L}{63}\right).\] Divide both sides by \(-2.57\): \[\ln\left(\frac{87-L}{63}\right) = \frac{-4}{2.57} \approx -1.556.\] Find the exponent: \(e^{-1.556} \approx 0.211\). Set \(\frac{87-L}{63} = 0.211\): \(87 - L = 0.211 \times 63\). Calculate: \(87-L \approx 13.293\), thus \(L \approx 87 - 13.293 = 73.707\). So, the length is approximately 73.71 feet.

Step 3: Explaining Domain of the Formula

The domain of the function \(24 < L < 87\) is provided because the argument of the logarithm, \(\frac{87-L}{63}\), must be positive. If \(L\) is less than 87, the expression \(87-L\) is positive. However, if \(L\) is less than 24, the numerator \(87-L\) is greater than 63, leading to a fraction greater than 1; \(\ln(\text{positive number smaller than or equal to 1})\) is defined, but to ensure the model's applicability and consistency for this specific whale model, \(L\) should be greater than 24.

Key Concepts

Exponential Relationships

Exponential relationships are a fundamental part of understanding logarithmic functions and their applications in real-world scenarios. These relationships describe how one variable can change exponentially based on another variable. In the formula provided, \(t=-2.57 \ln\left(\frac{87-L}{63}\right)\), we see an example of an exponential relationship involving the age \(t\) of the blue whale as it relates to its length \(L\).

The logarithmic function \(\ln(x)\) is the inverse of the exponential function \(e^x\). This means that in contexts like age estimation or population growth, logarithms can "decode" the exponential growth patterns, turning them into manageable calculations. Exponential relationships often appear:

• In financial calculations, like compound interest
• In biological contexts, like cell growth or decay
• In physical phenomena, like radioactive decay

By understanding these relationships, you can predict how changes in one quantity will affect another. In the case of our exercise, seeing \(t = -2.57 \ln\left(\frac{87-L}{63}\right)\), we understand that even small changes in the length of the blue whale can considerably affect its estimated age, highlighting the power of exponential relationships.

Domain and Range

To define a function properly, understanding its domain and range is crucial. For the function describing the blue whale's age, \(24 < L < 87\), the domain specifies the possible values that the variable \(L\) (length) can take.

The reason this domain is set, starts with the nature of the logarithm. The logarithm \(\ln(x)\) requires that its argument \(x\) is greater than zero because \(\ln(0)\) is undefined, and \(\ln(x)\) is only defined for positive numbers. In the formula \(\ln\left(\frac{87-L}{63}\right)\), the expression \(\frac{87-L}{63}\) needs to remain positive, which requires \(L < 87\).

Moreover, the condition \(L > 24\) ensures that \(\frac{87-L}{63}\) remains a usable fraction for this model. If \(L\) were less than 24, the argument could become greater than one, potentially making the output unpredictable within the model's context. Thus, by defining this specific domain, we can ensure that calculations remain consistent and meaningful.

• Domain sets input limits (e.g., length \(L\))
• Range specifies possible output values (e.g., age \(t\))

Understanding domain and range helps prevent errors and ensures a model's applicability for the situation being examined.

Age Estimation

Age estimation using mathematical models is a fascinating application of functions, particularly in biology and archaeology. By using equations like the one given for the blue whale, we can estimate a whale’s age based on measurable attributes like length.

The model \(t = -2.57 \ln\left(\frac{87-L}{63}\right)\) uses a logarithmic function to approximate the age \(t\). This is a practical method for non-invasive age estimation, which is especially important for conservation and study of wildlife.

• Age estimation can provide insights into life expectancy.
• It helps in understanding growth patterns of species.
• Allows researchers to monitor population health and dynamics.

In practice, scientists collect data on animal characteristics and then use mathematical models to predict other traits, such as age. By validating these models with actual samples where the age is known, the reliability of these estimates can be assessed. This process allows for effective wildlife management and helps in the strategic allocation of conservation efforts.