Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Stories are told about some of the less fair-minded teams of early baseball (e.g., the Baltimore Orioles of the 1890 s) freezing baseballs until shortly before game time so that although the cover would feel normal, the core of the ball would be much colder. Then, they would attempt to introduce these balls into play when the opposing team was at bat, working on the assumption that the frozen balls would not travel as far when hit. Experiments have shown that a ball whose temperature is \(-10^{\circ} \mathrm{F}\) would travel 350 feet after a given swing of the bat, while a ball whose temperature is \(150^{\circ} \mathrm{F}\) would be hit 400 feet by the same swing. Assume this relationship is linear. Let \(B(T)\) be the distance this swing would produce, where \(T\) is the temperature in degrees Fahrenheit. (a) Find an equation for \(B(T)\). (b) What is the \(B\) -intercept? What is its practical meaning? (c) What is the slope of \(B(T) ?\) What is its practical meaning?

Step-by-Step Solution

Answer

a) The equation for \(B(T)\) is \(B(T) = 0.3125T + 353.125\). b) The B-intercept is 353.125. This means the ball would travel 353.125 feet if the temperature were 0 degrees Fahrenheit. c) The slope of \(B(T)\) is 0.3125. This means the distance the ball would travel increases by 0.3125 feet for each degree increase in temperature.

Step 1: Determine the slope

First, two points from the given relationship are to be considered, \((-10, 350)\) and \((150, 400)\). The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \(m = (y_2 - y_1) / (x_2 - x_1)\). Substituting the coordinates of our points into the formula, we get \(m = (400-350) / (150 - (-10)) = 50 / 160 = 0.3125\).

Step 2: Apply the slope to the equation

Substitute the found slope \(m = 0.3125\) into the equation of a line in slope-intercept form \(y = mx + b\), giving \(B(T) = 0.3125T + b\).

Step 3: Find the B-intercept

Choose a point from the given pairs and substitute the values into the function to solve for \(b\). Using the point \((-10, 350)\) in our example, we substitute \(T = -10\) and \(B(T) = 350\) into our equation and solve for the unknown \(b\). Thus, we get: \(350 = 0.3125 * -10 + b\), which solves to give \(b = 353.125\). Consequently, the complete linear equation representing the relationship between temperature and the distance the ball travels when hit is \(B(T) = 0.3125T + 353.125\).

Step 4: Interpret the B-intercept and the Slope

The B-intercept, \(b = 353.125\), is the theoretical distance the ball would travel (in feet) if the temperature were 0 degrees Fahrenheit. The slope, \(m = 0.3125\), shows how much the distance the ball travels increases (in feet) for each degree increase in temperature.

Key Concepts

Finding Slope

In calculus and algebra, the slope of a line is a measure of how steep it is. It's calculated as the 'rise over run', which denotes how much the line goes up (or down) for a horizontal distance. Specifically, for two points on a linear graph, the slope is the change in the y-value divided by the change in the x-value.

To find the slope of a relationship represented by two points, you use the formula:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \).
In the context of our baseball problem, we calculated the slope using points \((-10, 350)\) and \((150, 400)\), where the first number in each pair is the temperature and the second is the distance the ball travels. The slope, representing the rate of change, tells us how much the hitting distance changes for each degree of change in temperature. In simpler terms, it's about understanding how temperature affects how far the ball will travel when hit.

Slope-Intercept Form

The slope-intercept form of a line is a straightforward way to write a linear equation. It's given by the equation \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. This form is useful because it clearly shows how a line's slope affects its position and how high or low it crosses the y-axis.

In our baseball scenario, by substituting in the calculated slope of 0.3125 and solving for \( b \), we determined the complete equation to represent the relationship in slope-intercept form as \( B(T) = 0.3125T + 353.125 \). This means that, for each additional degree Fahrenheit in temperature, the ball will, on average, travel an additional 0.3125 feet.

B-Intercept Meaning

The B-intercept, or the y-intercept, is the point where the line crosses the y-axis on a graph. In our linear equation \( B(T) \), the B-intercept is \( b = 353.125 \). This represents the scenario where the independent variable \( T \) is zero. Despite being a theoretical value when it comes to temperature affecting the baseball distance, it tells us the distance the ball would hypothetically travel if the temperature were 0°F. In practice, it gives us a baseline from which to compare how changes in temperature affect the ball's travel distance.

Effect of Temperature on Baseball Distance

The physical behavior of a baseball is influenced by temperature. In colder conditions, the baseball is less elastic and doesn't compress as much when hit, leading to shorter distances. Conversely, warmer conditions increase the ball's elasticity and, therefore, the distance it can travel. This linear relationship modeled by the function \( B(T) = 0.3125T + 353.125 \) shows that with each increase in temperature, the ball travels further.

Using the given data in our problem, the difference in travel distances for the ball at temperatures of \(-10^\circ F\) and \(150^\circ F\) was incorporated into a linear model. This allows us to predict how much farther the ball would theoretically travel at any given temperature within that range. The slope of 0.3125 indicates the amount of increase in feet per degree Fahrenheit, showing a tangible way that physics affects sports performance.