Problem 26 - Applied Calculus

Question

Table 1.8 gives the average weight, \(w,\) in pounds, of 20 American men in their sixties for height, \(h\), in inches. (a) How do you know that the data in this table could represent a linear function? (b) Find weight, \(w\), as a linear function of height, \(h\) What is the slope of the line? What are the units for the slope? (c) Find height, \(h,\) as a linear function of weight, \(w\) What is the slope of the line? What are the units for the slope? (d) What are the vertical and horizontal intercepts of the function? Give units and interpret the answers. $$\begin{array}{l|c|c|c|c|c|c|c|c}\hline h \text { (inches) } & 68 & 69 & 70 & 71 & 72 & 73 & 74 & 75 \\\\\hline w \text { (pounds) } & 166 & 171 & 176 & 181 & 186 & 191 & 196 & 201 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

Weight is a linear function of height with a slope of 5 pounds/inch. Height is a linear function of weight with a slope of 1/5 inches/pound.

Step 1: Recognizing a Linear Function

To determine if the data can be represented by a linear function, we should see if the increase in weight (\(w\)) is consistent for each inch increase in height (\(h\)). Looking at the table, each inch increase in height consistently results in a 5-pound increase in weight. This consistent rate of change indicates that weight is a linear function of height.

Step 2: Express Weight as a Linear Function of Height

Using the formula for a linear function, \(w = mh + b\), where \(m\) is the slope. We calculate the slope as follows:\[m = \frac{\Delta w}{\Delta h} = \frac{171 - 166}{69 - 68} = 5/1 = 5\]Thus, the slope (\(m\)) of the line is 5, and it represents the pounds per inch. To find the y-intercept (\(b\)), use a data point, for example, \(68, 166\):\[166 = 5(68) + b \rightarrow b = 166 - 340 = -174\]Therefore, the linear function is \(w = 5h - 174\).

Step 3: Express Height as a Linear Function of Weight

To write height as a linear function of weight, we use the inverse of the previous function \(w = 5h - 174\). Solving for \(h\):\[w = 5h - 174\]\[h = \frac{w + 174}{5}\].The slope of this function is \( \frac{1}{5} \), representing inches per pound.

Step 4: Determine Vertical and Horizontal Intercepts

To find the vertical intercept of \(w = 5h - 174\), set \(h = 0\):\[w = 5(0) - 174 = -174\]which is impractical in context. For the horizontal intercept, set \(w = 0\):\[0 = 5h - 174 \rightarrow h = \frac{174}{5} = 34.8\]Again, impractical as it implies a negative weight below tested range. Intercepts indicate where the function crosses axes.

Key Concepts

Slope

The slope is a key characteristic of a linear function. It describes how steep a line is. In mathematical terms, slope is the ratio of the change in the dependent variable (here, weight) to the change in the independent variable (height). For this exercise, when we talk about slope, we're referring to how much the weight of a person changes with each additional inch of height.
Looking at our table, we see that for each increase in height by one inch, weight increases by 5 pounds. This tells us the slope is 5. This value represents 'pounds per inch,' emphasizing the relationship between height and weight.
Understanding the slope is crucial since it helps in visualizing the direction and steepness of the linear function line on a graph.

Intercepts

Intercepts are the points where a graph crosses the axes. In our function, we have two types: vertical and horizontal intercepts.
The vertical intercept, or y-intercept, is at the point where the line crosses the y-axis, which happens when the height is zero. In our function, solving for height at zero gives us \[-174\] which is not practical in real terms, but mathematically, it represents the starting point of our function.
On the other hand, the horizontal intercept, or x-intercept, is where the line meets the x-axis, meaning the weight is zero. Solving for weight equals zero tells us that \[h = 34.8\]. Once again, though this doesn't fit our data range, it provides a complete view of the line extending on the plane. Recognizing these intercepts aids in understanding where the function changes form on a graph.

Height-Weight Relationship

The height-weight relationship is a great example of how linear functions manifest in real-world situations. In this case, we are looking at how a man's height in inches is correlated to his weight in pounds.
Our data show that as height increases, weight also rises at a constant rate. This consistent increase indicates a strong, predictable relationship - perfectly captured by the linear nature of the function.
Such relationships are valuable for making predictions. For instance, given a height not found in the table, you can use the linear equation we've formed to estimate the expected average weight. This predictive ability is essential in fields such as health and nutrition.

Linear Equation

A linear equation represents the relationship between two variables in a straight line when graphed. In our case, the linear equation is \[w = 5h - 174\]. This equation summarizes how weight (\(w\)) changes with height (\(h\)).
The form of a linear equation is usually given as \(y = mx + b\), where \(m\) is the slope and \(b\) is the intercept. In our specific form, weight is a function of height, showing that weight increases by 5 pounds when height increases by 1 inch.
Linear equations like ours are straightforward and useful in expressing relationships in data. They allow for easy computation of any one variable, given the other, and are widely applicable in diverse areas of study.