Problem 101 - Algebra and Trigonometry with Analytic Geometry

Question

The average weight \(W\) (in pounds) for men with height \(h\) between 64 and 79 inches can be approximated using the formula \(W=0.1166 h^{1.7}\). Construct a table for \(W\) by letting \(h=64,65, \ldots, 79\). Round all weights to the nearest pound. $$ \begin{array}{|c|c|c|c|} \hline \text { Height } & \text { Weight } & \text { Height } & \text { Weight } \\ \hline 64 & & 72 & \\ \hline 65 & & 73 & \\ \hline 66 & & 74 & \\ \hline 67 & & 75 & \\ \hline 68 & & 76 & \\ \hline 69 & & 77 & \\ \hline 70 & & 78 & \\ \hline 71 & & 79 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Weights: 136, 141, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 206, 211.

Step 1: Understand the formula

The formula given to approximate the weight based on height is \(W = 0.1166 h^{1.7}\), where \(W\) is the weight in pounds and \(h\) is the height in inches. We will use this to calculate the weights for each height from 64 to 79 inches.

Step 2: Calculate weights for each height

For each height \(h\) from 64 to 79, calculate the weight \(W\) using the formula \(W = 0.1166 h^{1.7}\). Perform the calculation for each height and maintain precision in your intermediate steps.

Step 3: Round the weights

For each calculated weight \(W\), round the result to the nearest whole number, as specified in the problem.

Step 4: Construct the table

Fill in the table with the heights in one column and the corresponding rounded weights in the adjacent column.

Key Concepts

Weight calculation

Understanding weight calculation using a mathematical formula is key in estimating weights for different heights. In our exercise, the formula given is \(W = 0.1166 \cdot h^{1.7}\), where \(W\) represents weight in pounds and \(h\) stands for height in inches.

To calculate weight using this formula, substitute each height value into \(h\), and solve using standard arithmetic and exponent rules. Start with the lowest height value, 64 inches, and proceed to the highest, 79 inches. This process allows for creating a range of weight estimates.

It is important to carry out calculations step-by-step and accurately, especially when dealing with exponents, to ensure precision before rounding.

Height-weight formula

The height-weight formula used in this exercise models the relationship between height and weight. Such formulas are crucial in various fields, like health and fitness, as they help predict outcomes based on given values.

In our scenario, the formula \(W = 0.1166 \cdot h^{1.7}\) shows how weight changes based on height. The exponent \(1.7\) suggests that weight increases at a rate that is not directly proportional to height, reflecting a non-linear relationship.

Understanding this exponent is essential: it implies that simply doubling height would not double the weight. Instead, the weight varies according to the exponent's influence.

• This highlights the complexity of physical measurements and the need for precise calculations.
• It shows how mathematical modeling aids in realistic representations of everyday phenomena.

Mathematical modeling

Mathematical modeling involves using formulas to represent real-world scenarios. In this case, our model predicts weight based on height, providing practical applications in fields like health assessments and ergonomics.

Models need to be simple yet accurate, which is why parameters like the coefficient \(0.1166\) and the exponent \(1.7\) are expertly chosen. They offer a balance between simplicity and reliability.

Mathematical models:

• Allow predictions based on given inputs, in our case, height values.
• Help in analyzing relationships and trends, identifying how factors like height affect body weight.

Building and understanding such models require insights into both the theoretical and practical aspects of the formula's components.

Rounding numbers

Rounding is the final step in our computation process, transforming exact figures into more digestible, user-friendly numbers. Typically, rounding is done to the nearest whole number, making it easier to communicate and use in daily applications.

In this exercise, once the weight \(W\) is calculated for a height \(h\), the result is rounded to the nearest pound.

• To round a number, check the digit following the desired decimal place.
• If this digit is 5 or higher, round the last desired digit up. If it's less than 5, leave it unchanged.

Rounding ensures calculated values maintain practicality without unnecessary precision, streamlining interpretations in tables or reports.