Problem 55 - Calculus

Question

The table gives the average weight, \(w,\) in pounds, of American men in their sixties for height, \(h,\) in inches. 10 (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?$$\begin{array}{l|r|r|r|r|r|r|r|r}\hline h \text { (inches) } & 68 & 69 & 70 & 71 & 72 & 73 & 74 & 75 \\\\\hline w \text { (pounds) } & 166 & 171 & 176 & 181 & 186 & 191 & 196 & 201 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

(a) Constant change in weight suggests linearity. (b) The slope is 5 pounds per inch. (c) The slope is 0.2 inches per pound.

Step 1: Determine Linear Function Possibility

To determine if the data can represent a linear function, we check if there is a constant rate of change in weight for each unit change in height. Calculate the difference in weight for each successive inch of height: 171 - 166 = 5, 176 - 171 = 5, 181 - 176 = 5, 186 - 181 = 5, 191 - 186 = 5, 196 - 191 = 5, 201 - 196 = 5. Since the change is constant, the data could represent a linear function.

Step 2: Express Weight as a Linear Function of Height

To express weight, \(w\), as a linear function of height, \(h\), identify the slope and the y-intercept. The slope, \(m\), is the constant rate of change from Step 1, which is 5. Use one data point (e.g., (68,166)) to find the y-intercept, \(b\), using the equation \(w = mh + b\): \(166 = 5 \times 68 + b\). This simplifies to \(b = 166 - 340 = -174\). Therefore, the function is: \(w = 5h - 174\).

Step 3: Slope Units for Weight Function

The slope of the line from Step 2 is 5, which represents a change in weight per change in height. Therefore, the units of the slope are pounds per inch.

Step 4: Express Height as a Linear Function of Weight

To find height \(h\) as a function of weight \(w\), we invert the linear function from Step 2. Start with \(w = 5h - 174\), solve for \(h\): \(5h = w + 174\), thus \(h = \frac{w + 174}{5}\), which simplifies to \(h = \frac{1}{5}w + \frac{174}{5}\). Therefore, \(h = 0.2w + 34.8\).

Step 5: Slope Units for Height Function

The slope of the line when expressing height as a function of weight is \(0.2\). This slope represents the change in height per unit change in weight, and the units are inches per pound.

Key Concepts

Slope

When discussing linear functions, the slope is one of the cornerstone concepts. The slope of a line defines its steepness, and it is determined by the ratio of the vertical change to the horizontal change between two distinct points on the line. For example, let's take a look at the problem previously discussed regarding the relationship between height and weight.Here, as height (\( h \)) increases by one inch, weight (\( w \)) consistently increases by 5 pounds. This constant change forms the slope of the line, which in this example is \(5\), demonstrating a direct relationship.

• The value of the slope indicates how much the dependent variable (weight) changes with the independent variable (height).
• A positive slope, like 5, means an increase in the weight as height increases.
• In this instance, the slope's units are pounds per inch, representing the additional weight gained per inch of height.

Understanding the slope is vital as it reveals the rate at which one variable affects another in a linear function.

Rate of Change

The rate of change in a linear function refers to how one variable changes in relation to another. It is essentially another way of understanding the slope. In a real-world situation, such as the height and weight of individuals, it tells us how much weight changes as height increases.

• In the problem, the rate of change is 5 pounds per inch. This constant indicates that each additional inch of height corresponds to an increase of 5 pounds in weight.
• This consistent rate is what makes the function linear; a constant increase or decrease signifies a direct correlation.
• Understanding the rate of change is essential for interpreting data trends and relationships clearly and effectively.

The rate of change helps us understand real-world phenomena, enabling predictions and analyses based on the linear relationship between variables.

Linear Equation

A linear equation represents a straight line when graphed on a coordinate plane. It can be represented in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The y-intercept is the value of \(y\) when \(x = 0\).In this context:- The weight (\(w\)) is expressed as a linear function of height (\(h\)): \(w = 5h - 174\).- The height is expressed as a function of weight: \(h = 0.2w + 34.8\).

• The slope (\(m\)), whether 5 or 0.2, depicts the relationship's strength and direction.
• The y-intercept (\(-174\) or \(34.8\)) tells us where the line crosses the axis, which is useful for contextualizing data when measurements start from zero.
• Equations like these are central to modeling relationships and predicting values in various situations.

Linear equations provide a straightforward way to anticipate changes and trends in related data, making them indispensable in fields ranging from physics to economics.