Problem 65 - Algebra and Trigonometry with Analytic Geometry

Question

The Ehrenberg relation $$ \ln W=\ln 2.4+(1.84) h $$ is an empirically based formula relating the height \(h\) (in meters) to the average weight \(W\) (in kilograms) for children 5 through 13 years old. (a) Express \(W\) as a function of \(h\) that does not contain \(\ln\). (b) Estimate the average weight of an 8-year-old child who is \(1.5\) meters tall.

Step-by-Step Solution

Answer

The average weight of a 1.5 meter tall 8-year-old is approximately 37.85 kg.

Step 1: Isolate W in the Equation

The given equation is \( \ln W = \ln 2.4 + 1.84h \). To express \( W \) as a function of \( h \), we need to get rid of the natural logarithm. We can do this by exponentiating both sides of the equation to solve for \( W \). This leads to \( W = e^{\ln 2.4 + 1.84h} \).

Step 2: Simplify the Exponentiation

Using the properties of exponents: \( e^{a+b} = e^a \cdot e^b \), we rewrite the equation as \( W = e^{\ln 2.4} \cdot e^{1.84h} \). Since \( e^{\ln 2.4} = 2.4 \), the expression simplifies to \( W = 2.4 \cdot e^{1.84h} \). Thus, \( W(h) = 2.4 \cdot e^{1.84h} \).

Step 3: Substitute h = 1.5 to Estimate W

Now we will estimate the average weight of an 8-year-old child who is 1.5 meters tall by substituting \( h = 1.5 \) into the function \( W(h) = 2.4 \cdot e^{1.84h} \). This gives us \( W(1.5) = 2.4 \cdot e^{1.84 \times 1.5} \).

Step 4: Calculate the Exponential Term

Calculate \( 1.84 \times 1.5 = 2.76 \). Therefore, \( W(1.5) = 2.4 \cdot e^{2.76} \). Use a calculator to find \( e^{2.76} \).

Step 5: Final Calculation and Result

Using a calculator, \( e^{2.76} \approx 15.77 \). Therefore, \( W = 2.4 \times 15.77 \approx 37.85 \). The estimated average weight of an 8-year-old child who is 1.5 meters tall is approximately 37.85 kilograms.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that raises a number, known as the base, to the power of an exponent. In the Ehrenberg relation, exponentiation is crucial for transforming expressions involving logarithms. When working with the equation \( \ln W = \ln 2.4 + 1.84h \), exponentiating each side helps remove the natural logarithm, simplifying our work.

• Exponentiation essentially "undoes" logarithms. This principle transforms log equations into more manageable forms.
• The formula \( e^{a+b} = e^a \cdot e^b \) is an essential property that allows decomposition of exponents, as used in this problem.
• Knowing that \( e^{\ln x} = x \), helps simplify expressions by canceling the logarithm with the exponential function.

Using these properties, the Ehrenberg relation becomes \( W = 2.4 \cdot e^{1.84h} \). This expression links height directly to an average weight, free from logarithmic terms.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). It often arises in natural growth patterns and exponential decay.

• The natural logarithm allows for the expression of relationships, like those in the Ehrenberg relation, in a linear form before converting.
• Logarithms provide a powerful way to simplify multiplication into addition, which was the initial form of the relation: \( \ln W = \ln 2.4 + 1.84h \).
• In many scientific formulas, logarithms are used to linearize exponential dynamics, which makes them easier to manipulate mathematically.

Understanding the properties of \( \ln \), like how it interacts with exponentials (\( e^{\ln x} = x \)), is critical for interpreting and transforming equations such as the Ehrenberg relation.

Function Notation

Function notation is a method to denote a function's output concerning its input variables, often written as \( f(x) \) or in this case, \( W(h) \). It’s a concise way to represent the relationship between dependent and independent variables.

• Here, \( W(h) = 2.4 \cdot e^{1.84h} \) denotes the weight \( W \) as a function of height \( h \), clearly showing we derive \( W\) from \( h \).
• Function notation conveys the dependency and relationship inside our equation, making it explicitly clear what value depends on what variable.
• This notation is convenient for substituting numeric values to calculate specific results, such as predicting the weight for given heights, like the 1.5-meter child example.

So, when using \( W(h) \), it's all about clarity. Function notation tells you instantly what the variable of interest is and how it influences the result.