Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A horseman has some ponies of his own and boards horses for other people. For his own ponies, he orders 9 bales of hay from the supplier. The total number of bales he orders increases linearly with the number of horses he boards. When he boards 6 horses, he orders a total of 36 bales of hay (for these horses and his ponies). Express the number of bales of hay he orders as a function of the number of horses he boards.

Step-by-Step Solution

Answer

The number of bales of hay ordered as a function of the number of horses he boards is given by the formula \(f(x) = 4.5x + 9\).

Step 1: Determine the constant of the function

The constant is the point where the function crosses the y-axis; i.e. the bare minimum of bales the man orders no matter the number of horses. Since the man always orders 9 bales for his ponies no matter what, this is the constant of our function. Hence, the constant bale amount, let's call it 'b', equals 9.

Step 2: Determine the slope of the function

The slope is the amount of bales the man orders per horse. When 6 horses are boarded, the total amount of ordered bales is 36, including the 9 for his ponies. Therefore, the extra bales he ordered for the boarded horses is \(36 - 9 = 27\) bales. As these are for 6 horses, the amount per horse, let's call it 'm', is \(27 \div 6 = 4.5\) bales.

Step 3: Formulate the linear function

A linear function is typically written in the form \(f(x) = mx + b\), where \(x\) represents the variable (the number of boarded horses, in this case), \(m\) the slope, and \(b\) the constant. By plugging in our earlier values for 'm' and 'b', we find that \(f(x) = 4.5x + 9\).

Key Concepts

Slope of a Function

Imagine you're looking at a hill; the steepness of that hill can be thought of as the slope in mathematical terms. In a function, especially a linear one, the slope determines how fast the function changes as the independent variable, often denoted as 'x', changes. It is the ratio of the vertical change, or 'rise', to the horizontal change, or 'run'.

For the horseman's situation, we calculate the slope ('m') by finding out how many extra bales of hay are needed for each additional horse that's boarded. We derived this slope as 4.5 bales per horse, indicating that for every horse he takes in, the total number of hay bales he needs increases by 4.5. This concept is critical because once we know the slope, we can predict the number of bales needed for any number of horses. The mathematical expression for the slope is usually written as \( m = \frac{{rise}}{{run}} \) or in the context of our problem, \( m = \frac{{\text{{extra bales ordered}}}}{{\text{{number of boarded horses}}}} \).

Y-intercept

While slope tells us how quickly the number of hay bales increases, the y-intercept represents the starting point of the function when the number of boarded horses is zero. It is where the line representing the linear function crosses the y-axis on a graph.

For our horseman, the y-intercept is the number of hay bales he orders for just his own ponies, regardless of the boarding service. As calculated, this base number is 9 bales. Mathematically, we denote the y-intercept as 'b' in the linear equation \( y = mx + b \), where 'y' could represent the total bales of hay. When he boards no horses (when \( x = 0 \) ), the horseman will still order 9 bales, which means the graph of this function would intersect the y-axis at the point (0, 9). This highlights the importance of the y-intercept—it provides a baseline from which the slope 'm' operates.

Linear Equations

At the heart of our problem is a linear equation, which describes a relationship where one variable changes at a constant rate with respect to another. In simple terms, we're talking about a straight-line relationship. Any linear equation can be written in the form \( y = mx + b \), where 'y' is the dependent variable, 'm' is the slope, 'x' is the independent variable, and 'b' is the y-intercept.

In the context of the horseman's scenario, the linear equation we've crafted \( f(x) = 4.5x + 9 \) beautifully expresses how the total number of hay bales needed ('f(x)') changes linearly with each additional boarded horse ('x'). The power of this linear equation lies in its predictability allowing the horseman to calculate the required hay for any number of boarded horses quickly.

Function Representation

Functions come in various forms and representations, and understanding these can help easily grasp complex concepts. In our example, the function \( f(x) = 4.5x + 9 \) can be represented in several ways: as an equation, a table, a graph, and in words. Each representation serves its purpose and can offer insights into the behavior of the function.

As an equation, it provides a formula for calculating the total hay bales. If graphed, we'd see a straight line ascending from the point (0, 9), which portrays both the base order and how additional boarding affects it. In a table, you could list values of 'x' and the corresponding 'f(x)' providing a quick reference. Lastly, in words, it translates to 'The total hay bales are 9 plus 4.5 times the number of boarded horses.' This versatility in representation helps students connect different mathematical concepts and better understand their applications.