Problem 104 - A Graphical Approach to Precalculus with Limits

Question

Height and Weight 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 $$ w=7.46 h-374 $$ $$ \text { and } \quad w=7.93 h-405 $$ 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 tall. (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 (first model), 253 pounds (second model). (b) 65.96 inches. (c) 7.46 pounds (first model), 7.93 pounds (second model).

Step 1: Convert Height to Inches

We need to convert the player's height from feet and inches to inches only. Since 1 foot is 12 inches, a height of 6 feet 11 inches equals \(6 \times 12 + 11 = 72 + 11 = 83\) inches.

Step 2: Predict Weight Using First Model

Using the equation \( w = 7.46h - 374 \), substitute \( h = 83 \): \[ w = 7.46 \times 83 - 374 \]. Calculate the weight: \( w = 619.18 - 374 = 245.18 \). Round this to the nearest pound: \( 245 \) pounds.

Step 3: Predict Weight Using Second Model

Using the equation \( w = 7.93h - 405 \), substitute \( h = 83 \): \[ w = 7.93 \times 83 - 405 \]. Calculate the weight: \( w = 658.19 - 405 = 253.19 \). Round this to the nearest pound: \( 253 \) pounds.

Step 4: Solve for Height where Models Equal

Set the two weight equations equal to each other: \( 7.46h - 374 = 7.93h - 405 \). Solve for \( h \): \( 7.46h - 7.93h = -405 + 374 \), which simplifies to \( -0.47h = -31 \). Thus, \( h = \frac{-31}{-0.47} \approx 65.96 \) inches.

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

The change in weight with respect to height is the coefficient of \( h \) in the first model: \( 7.46 \) pounds per inch.

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

The change in weight with respect to height is the coefficient of \( h \) in the second model: \( 7.93 \) pounds per inch.

Key Concepts

Graphical Solutions

When tackling problems involving models, graphical solutions can be incredibly insightful. By plotting both equations, you can visually determine the relationship between variables. For our basketball player modeling:

• The x-axis represents height in inches (\( h \)), while the y-axis represents weight in pounds (\( w \)).
• By graphing the lines for both equations, you'll be able to see where they intersect.

The point where the lines intersect indicates the height at which the two models predict the same weight. This intersection is crucial as it visually confirms that no matter which model you use, at this specific height, the predicted weights coincide. Graphical solutions offer a visual testament to algebraic findings. This dual approach can increase comprehension by providing both a numeric and spatial representation of the problem.

Problem Solving

Mastering problem solving often means breaking a problem into more manageable parts. For this task:

• Start by converting any given measurements into a consistent unit. In this case, converting feet into inches ensures clarity.
• Apply the given formulas step by step to compute necessary values, such as predicting the weight of a player.

Next, solving when two models are equal involves:

• Setting both equations equal to find common values.
• Simplifying the equation for a straightforward solution for height or any variable of interest.

This exercise illustrates that, by systematically addressing each part of a problem while using algebraic manipulation, you can efficiently arrive at a solution without being overwhelmed.

Slope Interpretation

The slope of a line in an equation lets us understand the rate of change between the variables. In our context:

• The slope indicates how much the weight changes with a change in height.
• For the equation \( w = 7.46h - 374 \), the slope 7.46 tells us that for each inch increase in height, weight increases by 7.46 pounds.
• Similarly, the equation \( w = 7.93h - 405 \) has a slope of 7.93, which means each inch increase in height leads to a 7.93-pound increase in weight.

Understanding slopes helps infer the dependency between the variables and how sensitive one variable is to changes in another. This concept aids in real-world predictions, helping to choose a model that closely aligns with observed trends.