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Chapter 2: Problem 18

The women's recommended weight formula from Harvard Pilgrim Healthcare says, "Give yourself \(100 \mathrm{lb}\) for the first 5 ft plus 5 lb for every inch over 5 ft tall." a. Find a mathematical model for this relationship. Be sure you clearly identify your variables. b. Specify a reasonable domain for the function and then graph the function. c. Use your model to calculate the recommended weight for a woman 5 feet, 4 inches tall; and for one 5 feet, 8 inches tall.

Short Answer

Expert verified
The mathematical model is \( W = 5h - 200 \). The recommended weights are 120 lbs for a woman 5 feet 4 inches tall and 140 lbs for a woman 5 feet 8 inches tall.

Step by step solution

01

Identify the Variables

Let \(W\) represent the weight in pounds and \(h\) represent the height in inches.
02

Establish a Mathematical Model

The formula given states, '100 lbs for the first 5 feet plus 5 lbs for every inch over 5 feet.' Since there are 12 inches in a foot, 5 feet is equal to 60 inches. For heights above 60 inches, the weight increases by 5 lbs for each additional inch. Therefore, the equation can be written as: \[ W = 100 + 5(h - 60) \].
03

Simplify the Equation

Simplify the equation by distributing and combining like terms: \[ W = 100 + 5h - 300 \]. This simplifies to \[ W = 5h - 200 \].
04

Specify a Reasonable Domain

A reasonable domain for the function would be heights for adult women, which typically range from 4 feet 10 inches to 6 feet 2 inches. Converting to inches, this gives a domain of \(58 \leq h \leq 74\).
05

Graph the Function

Plot the function \( W = 5h - 200 \) on a graph, with \(h\) (height in inches) on the x-axis and \(W\) (weight in pounds) on the y-axis. Use the domain \(58 \leq h \leq 74\) and mark points at these boundaries as well as intermediate points like 60, 64, and 68 inches.
06

Calculate the Recommended Weight for 5 ft 4 in

First, convert 5 feet 4 inches to inches: \(5 \times 12 + 4 = 64 \) inches. Substitute \( h = 64 \) into the equation: \[ W = 5(64) - 200 = 320 - 200 = 120 \] lbs.
07

Calculate the Recommended Weight for 5 ft 8 in

First, convert 5 feet 8 inches to inches: \(5 \times 12 + 8 = 68 \) inches. Substitute \( h = 68 \) into the equation: \[ W = 5(68) - 200 = 340 - 200 = 140 \] lbs.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

mathematical modeling
Mathematical modeling helps us create equations or formulas based on real-world situations. In this exercise, we start by identifying the variables. Here, we let W represent weight in pounds, and h represent height in inches. By analyzing the provided formula, '100 lbs for the first 5 feet plus 5 lbs for every inch over 5 feet,' we establish a mathematical model to represent this relationship. Since 5 feet equals 60 inches, the formula translates to: \[ W = 100 + 5(h - 60) \]. By simplifying, we get \[ W = 5h - 200 \]. This equation allows anyone to calculate the recommended weight given a woman's height. Mathematical modeling is crucial, as it helps simplify complex relationships and make predictions.
domain and range
The domain of a function represents all possible input values. In our case, it's the range of heights in inches we want to consider for adult women. Typically, adult women's heights range from 4 feet 10 inches to 6 feet 2 inches. Converting these to inches, we determine the domain to be \(58 \leq h \leq 74\). The range, on the other hand, represents all possible output values, in this case, the calculated weights for these heights. Using our model, \( W = 5h - 200 \), we substitute the minimum height, 58 inches, and the maximum height, 74 inches, to find the range of weights. These calculations help us understand the limits within which our model is valid and applicable. Being precise with domain and range ensures that we do not use our model for values it's not designed to handle.
function graphing
Graphing a function visually represents the relationship between the variables. For our model, \( W = 5h - 200 \), we plot height (h) on the x-axis and weight (W) on the y-axis. Using the domain \(58 \leq h \leq 74\), we plot points for key values like 58, 60, 64, 68, and 74 inches. A linear function like this one forms a straight line. By graphing, we can easily see how weight increases with height and verify the model's accuracy. Additionally, graphing helps spot any anomalies and makes it easier to understand how changing one variable (height) affects the other (weight). Visual tools like graphs are essential, making abstract concepts concrete and comprehensible.

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Most popular questions from this chapter

The accompanying data show rounded average values for blood alcohol concentration \((\mathrm{BAC})\) for people of different weights, according to how many drinks ( 5 oz wine, 1.25 oz 80 -proof liquor, or 12 oz beer) they have consumed. $$ \begin{array}{cccc} \hline \text { Number of Drinks } & 100 \mathrm{lb} & 140 \mathrm{lb} & 180 \mathrm{lb} \\ \hline 2 & 0.075 & 0.054 & 0.042 \\ 4 & 0.150 & 0.107 & 0.083 \\ 6 & 0.225 & 0.161 & 0.125 \\ 8 & 0.300 & 0.214 & 0.167 \\ 10 & 0.375 & 0.268 & 0.208 \\ \hline \end{array} $$ a. Examine the data on BAC for a 100 -pound person. Are the data linear? If so, find a formula to express blood alcohol concentration, \(A,\) as a function of the number of drinks, \(D,\) for a 100 -pound person. b. Examine the data on BAC for a 140 -pound person. Are the data linear? If they're not precisely linear, what might be a reasonable estimate for the average rate of change of blood alcohol concentration, \(A,\) with respect to number of drinks, \(D ?\) Find a formula to estimate blood alcohol concentration, \(A,\) as a function of number of drinks, \(D,\) for a 140 -pound person. Can you make any general conclusions about BAC as a function of number of drinks for all of the weight categories? c. Examine the data on \(\mathrm{BAC}\) for people who consume two drinks. Are the data linear? If so, find a formula to express blood alcohol concentration, \(A,\) as a function of weight, \(W,\) for people who consume two drinks. Can you make any general conclusions about \(\mathrm{BAC}\) as a function of weight for any particular number of drinks?

Your bank charges you a \(\$ 2.50\) monthly maintenance fee on your checking account and an additional \(\$ 0.10\) for each check you cash. Write an equation to describe your monthly checking account costs.

Match each equation with the appropriate table. a. \(y=3 x+2\) b. \(y=\frac{1}{2} x+2\) c. \(y=1.5 x+2\) $$ \begin{aligned} &\text { A. } .\\\ &\begin{array}{ll} \hline x & y \\ \hline 0 & 2 \\ 2 & 3 \\ 4 & 4 \\ 6 & 5 \\ 8 & 6 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { B. }\\\ &\begin{array}{rr} \hline x & y \\ \hline 0 & 2 \\ 2 & 5 \\ 4 & 8 \\ 6 & 11 \\ 8 & 14 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { C. }\\\ &\begin{array}{rr} \hline x & \multicolumn{1}{c} {y} \\ \hline 0 & 2 \\ 2 & 8 \\ 4 & 14 \\ 6 & 20 \\ 8 & 26 \\ \hline \end{array} \end{aligned} $$

Determine which of the following tables represents a linear function. If it is linear, write the equation for the linear function. a. $$ \begin{array}{rr} \hline x & y \\ \hline 0 & 3 \\ 1 & 8 \\ 2 & 13 \\ 3 & 18 \\ 4 & 23 \\ \hline \end{array} $$ b.$$ \begin{array}{rr} \hline q & R \\ \hline 0 & 0.0 \\ 1 & 2.5 \\ 2 & 5.0 \\ 3 & 7.5 \\ 4 & 10.0 \\ \hline \end{array} $$ c. $$ \begin{array}{rr} \hline x & g(x) \\ \hline 0 & 0 \\ 1 & 1 \\ 2 & 4 \\ 3 & 9 \\ 4 & 16 \\ \hline \end{array} $$ d.$$ \begin{array}{cc} \hline t & r \\ \hline 10 & 5.00 \\ 20 & 2.50 \\ 30 & 1.67 \\ 40 & 1.25 \\ 50 & 1.00 \\ \hline \end{array} $$ e. $$ \begin{array}{rr} \hline x & \multicolumn{1}{c} {h(x)} \\ \hline 20 & 20 \\ 40 & -60 \\ 60 & -140 \\ 80 & -220 \\ 100 & -300 \\ \hline \end{array} $$ f. $$ \begin{array}{cc} \hline p & T \\ \hline 5 & 0.25 \\ 10 & 0.50 \\ 15 & 0.75 \\ 20 & 1.00 \\ 25 & 1.25 \\ \hline \end{array} $$

A hiker can walk 2 miles in 45 minutes. a. What is his average speed in miles per hour? b. What formula can be used to find the distance traveled, \(d,\) in \(t\) hours?
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