Problem 102 - Algebra and Trigonometry with Analytic Geometry

Question

The average weight \(W\) (in pounds) for women with height \(h\) between 60 and 75 inches can be approximated using the formula \(W=0.1049 h^{1.7}\). Construct a table for \(W\) by letting \(h=60,61, \ldots, 75\). Round all weights to the nearest pound. $$ \begin{array}{|c|c|c|c|} \hline \text { Height } & \text { Weight } & \text { Height } & \text { Weight } \\ \hline 60 & & 68 & \\ \hline 61 & & 69 & \\ \hline 62 & & 70 & \\ \hline 63 & & 71 & \\ \hline 64 & & 72 & \\ \hline 65 & & 73 & \\ \hline 66 & & 74 & \\ \hline 67 & & 75 & \\ \hline \end{array} $$ We sometimes use the notation and terminology of sets to describe mathematical relationships. A set is a collection of objects of some type, and the objects are called elements of the set. Capital letters \(R, S, T, \ldots\) are often used to denote sets, and lowercase letters \(a, b, x, y, \ldots\) usually represent elements of sets. Throughout this book, \(\mathbb{R}\) denotes the set of real numbers and \(\mathbb{Z}\) denotes the set of integers. Two sets \(S\) and \(T\) are equal, denoted by \(S=T\), if \(S\) and \(T\) contain exactly the same elements. We write \(S \neq T\) if \(S\) and \(T\) are not equal. Additional notation and terminology are listed in the following chart. $$ \begin{array}{|l|l|l|} \hline \begin{array}{c} \text { Notation or } \\ \text { terminology } \end{array} & \multicolumn{1}{|c|}{\text { Meaning }} & \multicolumn{1}{c|}{\text { Illustrations }} \\ \hline a \in S & a \text { is an element of } S & 3 \in \mathbb{Z} \\ a \notin S & a \text { is not an element of } S & \frac{3}{5} \notin \mathbb{Z} \\ \hline S \text { is a subset of } T & \begin{array}{l} \text { Every element of } S \text { is } \\ \text { an element of } T \end{array} & \mathbb{Z} \text { is a subset of } \mathbb{R} \\ \hline \text { Constant } & \begin{array}{l} \text { A letter or symbol that } \\ \text { represents a specific } \\ \text { element of a set } \end{array} & 5,-\sqrt{2}, \pi \\ \hline \text { Variable } & \begin{array}{l} \text { A letter or symbol that } \\ \text { represents any element } \\ \text { of a set } \end{array} & \begin{array}{l} \text { Let } x \text { denote any } \\ \text { real number } \end{array} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Weights range from 118 to 203 pounds for heights 60 to 75 inches.

Step 1: Understand the Formula

The formula given is \( W = 0.1049 \cdot h^{1.7} \), where \( W \) is the weight in pounds and \( h \) is the height in inches. Our goal is to calculate \( W \) for each height between 60 and 75 inches.

Step 2: Calculate for Each Height

For each height \( h \) from 60 to 75, substitute \( h \) into the formula and compute \( W \). Once calculated, round \( W \) to the nearest whole number.

Step 3: Create the Table

Record each height \( h \) along with its calculated weight \( W \) in a table format. This will allow us to systematically observe the relationship between height and weight according to the formula. Here is the completed table:\[ \begin{array}{|c|c|c|c|}\hline \text{Height} & \text{Weight} & \text{Height} & \text{Weight} \\hline 60 & 118 & 68 & 158 \\hline 61 & 123 & 69 & 164 \\hline 62 & 128 & 70 & 170 \\hline 63 & 133 & 71 & 176 \\hline 64 & 138 & 72 & 183 \\hline 65 & 144 & 73 & 190 \\hline 66 & 149 & 74 & 196 \\hline 67 & 154 & 75 & 203 \\hline\end{array} \]

Key Concepts

Sets and Set Notation

In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. Think of sets as simple containers holding certain elements. Elements can be anything: numbers, objects, or even other sets. There are specific symbols and terminology we use to describe relationships within sets. For instance:

• The notation \( a \in S \) means that \( a \) is an element of the set \( S \). For example, if we say \( 3 \in \mathbb{Z} \), it means the number 3 is part of the set of integers.
• On the contrary, \( a otin S \) means \( a \) is not a member of \( S \), like \( \frac{3}{5} otin \mathbb{Z} \), since \( \frac{3}{5} \) is a fraction, not an integer.
• When every element of one set \( S \) is also in a second set \( T \), we say \( S \) is a subset of \( T \) and write \( S \subseteq T \). For example, \( \mathbb{Z} \) (integers) is a subset of \( \mathbb{R} \) (real numbers).

Understanding sets and set notation provides a foundation for organizing and manipulating data mathematically. It's akin to learning the alphabet of mathematical language.

Exponents

Exponents are a concise way to express repeated multiplication of the same number or variable. When we see something like \( h^{1.7} \), it tells us the base \( h \) is multiplied by itself \( 1.7 \) times. This is particularly useful in mathematical modeling where the relationship between variables isn't straightforward.

Some key points about exponents include:

• \( h^1 \) means \( h \) to the power of one, which is simply \( h \).
• Fractional exponents, like in our problem, mean we are dealing with roots as well as powers: \( h^{1.7} \) indicates a complex relationship often not reducible to basic roots.
• Exponents follow certain rules such as the product of powers rule, \( a^m \times a^n = a^{m+n} \), and the power of a power rule, \((a^m)^n = a^{m \times n} \), among others.

These rules enable us to simplify expressions and solve equations more effectively. Working with fractional exponents, as seen in our weight formula, helps in creating more accurate models in scientific and practical applications.

Tabular Data Representation

Tabular data representation involves organizing data into rows and columns for clarity and efficient communication. Tables serve as a powerful tool, especially when we have a sequence of calculations or values that are interconnected. The table in our problem illustrates this by outlining heights and their corresponding weights calculated from the given formula.

Here’s why tables are helpful:

• They make it easy to spot trends, patterns, or errors at a glance.
• Tables organize data systematically, enhancing statistical analysis.
• They are excellent for comparing sets of values or demonstrating relationships between two or more variables, like how height relates to weight in the given exercise.

When constructing a table, always label your columns properly to avoid any confusion. For the given exercise, making a table helps visualize the relationship defined by the mathematical model, which strengthens understanding and ensures all data is accounted for accurately.