Problem 101 - Precalculus: Functions and Graphs

Question

The average weight \(W\) (in pounds) for men with height \(h\) between 64 and 79 inches can be approximated using the formula \(W=0.1166 h^{17}\). Construct a table for \(W\) by letting \(h=64,65, \ldots, 79 .\) Round all weights to the nearest pound. (TABLE CANT COPY)

Step-by-Step Solution

Answer

Create a table calculating \( W \) using the formula for heights 64 to 79 and round results.

Step 1: Understand the Formula

The given formula for the average weight, \( W \), in pounds based on height, \( h \), is \( W = 0.1166 h^{17} \). This formula is used to calculate the average weight for each height from 64 to 79 inches.

Step 2: Set up the Table

We will create a table with two columns: one for height \( h \) and the other for weight \( W \). The heights will range from 64 to 79 inches, and for each height, we will calculate the corresponding weight using the formula.

Step 3: Calculate Weight for Each Height

Using the formula \( W = 0.1166 h^{17} \), calculate \( W \) for each height from 64 to 79. For example, for \( h = 64 \), plug it into the formula to get \( W = 0.1166 \times 64^{17} \). Repeat for each value of \( h \).

Step 4: Round the Weights

Round the calculated weight \( W \) for each height to the nearest pound. This will make the values more practical and easier to read.

Step 5: Complete the Table

Fill in the table with each height from 64 to 79 and the corresponding rounded weight. Ensure each computed and rounded weight is aligned correctly with its height.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable power. They are commonly expressed in the form \( f(x) = a \, b^x \), where \( a \) is a constant, \( b \) is the base, and \( x \) is the exponent. In this exercise, we use an exponential function to model the relationship between height and weight. The formula given is \( W = 0.1166 \, h^{17} \). Here, \( h \) is the height serving as the base, and 17 is the exponent. This characterizes an exponential relationship where small changes in \( h \) can result in significant changes in \( W \) due to the large exponent.Exponential functions like this can model various real-world phenomena where growth or decay occurs at increasing rates, such as population growth, radioactive decay, and, as seen here, certain biological relationships.

Weight Calculation

Calculating weight using the given formula involves substituting the height value into the equation. For each specific height, we compute the weight using the formula \( W = 0.1166 \, h^{17} \).The process of weight calculation is detailed as follows:

• Identify the height \( h \) value.
• Raise the height \( h \) to the 17th power.
• Multiply the result by the constant 0.1166.

This gives us the calculated weight before rounding. By performing these calculations for each height in the specified range, we generate a series of weight values that reflect the mathematical model used in the problem.

Height-Weight Relationship

The height-weight relationship described in this exercise highlights how mathematical equations can describe biological attributes. For men, the average weight tends to increase significantly with an increase in height, following the model given by the formula \( W = 0.1166 \, h^{17} \).This is not a straightforward proportional relationship because the exponent involves a power of 17, which causes the weight to grow much faster than the linear increase in height. Consider these factors in any height-weight relationship:

• The base (height) and its impact when raised to a power.
• The role of constants such as 0.1166 in scaling the outcome.
• The non-linear nature interpreted through exponential growth.

Understanding this relationship aids in comprehending how even small increases in height can result in large variations in weight prediction when modeled exponentially.

Rounding Numbers

Rounding numbers is a crucial step in this exercise to make the calculated weights more applicable and user-friendly. The process of rounding ensures precision while simplifying the result, which aids in easier interpretation. Here's how you can round numbers effectively:

• Determine the place to which you are rounding, such as the nearest pound.
• Select a calculated weight, for instance, 243.6 pounds.
• If the digit after the rounding place is 5 or above, round up.
• If it is lower than 5, round down.

For example, rounding 243.6 results in 244, as 6 is greater than 5. This method helps present data more clearly without significantly altering the underlying information. Rounding is essential when modeling real-world phenomena to ensure that data remains practical and manageable.