Problem 90 - A Graphical Approach to College Algebra

Question

The relationship between a professional basketball player's height \(h\) in inches and weight \(w\) in pounds was modeled by using two samples of players. The resulting equations were $$\begin{aligned}&w=7.46 h-374\\\&w=7.93 h-405\end{aligned}$$ and Assume that \(65 \leq h \leq 85\) (a) Use each equation to predict the weight to the nearest pound of a professional basketball player who is 6 feet 11 inches. (b) Determine graphically the height at which the two models give the same weight. (c) For each model, what change in weight is associated with a 1 -inch increase in height?

Step-by-Step Solution

Answer

(a) 245 pounds, 253 pounds; (b) 66 inches; (c) 7.46 pounds, 7.93 pounds, respectively.

Step 1: Convert Height into Inches

First, convert the basketball player's height from feet and inches into inches. 6 feet 11 inches equals \( 72 + 11 = 83 \) inches.

Step 2: Use First Equation to Predict Weight

Substitute \( h = 83 \) into the first equation \( w = 7.46h - 374 \):\[w = 7.46(83) - 374\]Calculate the weight:\[w = 619.18 - 374 = 245.18\]Thus, the predicted weight is approximately 245 pounds (rounded to the nearest pound).

Step 3: Use Second Equation to Predict Weight

Substitute \( h = 83 \) into the second equation \( w = 7.93h - 405 \):\[w = 7.93(83) - 405\]Calculate the weight:\[w = 658.19 - 405 = 253.19\]Thus, the predicted weight is approximately 253 pounds (rounded to the nearest pound).

Step 4: Determine Height Where Equations Are Equal

Set the two equations equal to find the height where weights match:\[7.46h - 374 = 7.93h - 405\]Rearrange to find \( h \):\[7.46h - 7.93h = -405 + 374\]\[-0.47h = -31\]\[h = \frac{31}{0.47} \approx 65.96\]Thus, the height at which both equations give the same weight is approximately 66 inches.

Step 5: Calculate Weight Change per Inch Increase for First Model

The first equation is \( w = 7.46h - 374 \). The coefficient of \( h \) (7.46) represents the rate of change in weight with respect to height. Therefore, for each additional inch, the weight increases by approximately 7.46 pounds according to the first model.

Step 6: Calculate Weight Change per Inch Increase for Second Model

The second equation is \( w = 7.93h - 405 \). Similarly, the coefficient of \( h \) (7.93) represents the rate of change in weight with respect to height. Therefore, for each additional inch, the weight increases by approximately 7.93 pounds according to the second model.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra and crucial for understanding relationships between variables. In this case, we have two linear equations relating a basketball player's height, \( h \), to their weight, \( w \). Linear equations take the form \( w = mh + b \), where \( m \) is the slope and \( b \) is the y-intercept. - **Slope (\( m \))**: This represents the change in weight per inch increase in height. In our equations, the slopes are 7.46 and 7.93. - A higher slope indicates a steeper increase in weight as height increases.- **Y-intercept (\( b \))**: This signifies the weight when the height is zero, though not practically useful here, it helps define the equation's line.Understanding how slopes and intercepts affect linear equations helps us predict outcomes, like predicting the weight of a player when you input their height into the equation. This is crucial for solving problems involving real-world relationships.

Graphical Analysis

Graphical analysis is a visual approach to understand the relationships modeled by equations. It involves plotting the linear equations on a graph to see how they compare.In this exercise, the two weight-height equations can be represented as straight lines on a coordinate plane:- **X-axis and Y-axis**: - Height \( h \) is plotted on the x-axis. - Weight \( w \) is plotted on the y-axis.Graphing these lines allows us to find where they intersect. The point of intersection represents a height where both models predict the same weight. To find this graphically:1. Draw both lines using their respective equations.2. Identify the point where the lines meet.For this exercise, that height is approximately 66 inches. Graphs provide a clear, visual method to compare equations and see where they lead to the same or different results.

Mathematical Modeling

Mathematical modeling involves using mathematical constructs, like equations, to represent real-world scenarios. It's a crucial skill in college algebra and beyond. In this exercise, the two given linear equations model the relationship between a basketball player's height and weight. - **Model Selection**: - Each model provides a slightly different prediction of weight based on height. - Choose a model based on the context or additional data if needed. - **Utility of Modeling**: - Models simplify complex relationships into understandable formats. - They allow us to predict unknown values, like weight for a specific height, using equations. Each model assumes a constant rate of change in weight per inch increase in height, but real-world data may vary. Therefore, modeling is not just about computation; it requires critical thinking to evaluate its accuracy and reliability in representing real-life conditions.