Problem 64 - Algebra A Combined Function

Question

On an NBA team the two forwards measure \(6^{\prime} 8^{\prime \prime}\) and \(6^{\prime} 6^{\prime \prime}\) tall and the two guards measure \(6^{\prime} 0^{\prime \prime}\) and \(5^{\prime} 9^{\prime \prime}\) tall. How tall should the center be if they wish to have a starting team average height of at least \(6^{\prime} 5^{\prime \prime} ?\)

Step-by-Step Solution

Answer

The center should be 7 feet 2 inches tall.

Step 1: Convert Heights to Inches

Convert each player's height into inches so that calculations are consistent. - First forward: \(6^{\prime} 8^{\prime\prime}\) = 6 feet * 12 + 8 = 80 inches.- Second forward: \(6^{\prime} 6^{\prime\prime}\) = 6 feet * 12 + 6 = 78 inches.- First guard: \(6^{\prime} 0^{\prime\prime}\) = 6 feet * 12 + 0 = 72 inches.- Second guard: \(5^{\prime} 9^{\prime\prime}\) = 5 feet * 12 + 9 = 69 inches.

Step 2: Convert Desired Average Height to Inches

The desired average height of the team is \(6^{\prime} 5^{\prime\prime}\), which is equivalent to 6 feet * 12 + 5 = 77 inches.

Step 3: Calculate Total Height Needed for Desired Average

To find the total height needed for a desired average of 77 inches among 5 players, calculate: 77 inches/player * 5 players = 385 inches.

Step 4: Calculate the Combined Height of Current Players

Add the heights of the two forwards and two guards: 80 + 78 + 72 + 69 = 299 inches.

Step 5: Determine Height Required for the Center

Subtract the combined height of the current players from the total height required for the average: 385 inches (total needed) - 299 inches (current total) = 86 inches.

Step 6: Convert Center's Height Back to Feet and Inches

Convert 86 inches to feet and inches: 86 inches = 7 feet * 12 + 2 = 7 feet 2 inches. Thus, the center needs to be 7 feet 2 inches tall.

Key Concepts

Converting Units

When dealing with measurements like height in different units, it's crucial to make everything uniform for accurate calculations. In this exercise, player heights are originally given in feet and inches. To simplify, we convert these measurements into a single unit: inches.
This is done by using the conversion factor that 1 foot equals 12 inches.

• For a height of 6 feet 8 inches, calculate by multiplying 6 by 12 (since each foot is 12 inches) and then add 8 for the inches, totaling 80 inches.
• Similarly, for 6 feet 6 inches, you do 6 times 12 plus 6, which gives you 78 inches.
• This makes all subsequent calculations much easier since you are dealing with additions and subtractions of uniform units.

Understanding and applying this conversion method ensures accurate mathematical processing of data, especially where precision is needed, such as calculating averages or totals.

NBA Team Statistics

In the sports world, especially in the NBA, player statistics like height are crucial for planning strategies and forming balanced teams.
Having average height statistics allows teams to arrange members in a way that exploits strengths and covers weaknesses.

• Each position on a basketball team, like forward, guard, and center, has suggested height ranges for optimal performance.
• In this example, the team wanted the average starting height to meet or exceed a certain standard (6 feet 5 inches) to ensure competitiveness on the court.

Knowing how to calculate this strategically influences decisions on team formations and player roles. It highlights the importance of using statistics to align with game objectives.

Mathematical Problem Solving

Solving the problem of how tall the center must be to achieve the desired average height illustrates a practical use of mathematical problem-solving skills.
This involves several key steps:

• First, understanding the problem requirements and what is being asked.
• Next, assembling and working through numerical data, such as player heights.
• It’s important to break the problem down into smaller manageable steps, like calculating total needed height or converting units as seen previously.

This scenario demonstrates the blend of logical reasoning and mathematical calculations to solve practical problems, a valuable skill in both academic and real-world scenarios.
Problems like this can help sharpen critical thinking and enhance numerical literacy.

Step-By-Step Solution

Approaching a problem using a structured step-by-step method is essential for clear thinking and ensures that every aspect of the problem is addressed comprehensively.
This method helps in both academic exercises and real-world applications:

• By meticulously following each step, like converting units to ensure consistent measurements or calculating totals to meet specific targets, mistakes are minimized.
• In this exercise, the process begins by converting units, calculating desired totals, and ultimately determining how the final component (the center's height) fits into the equation.

A structured solution path not only helps in finding the correct answer but also aids in understanding the reasoning behind each step.
This approach can be repeated in other contexts, helping students approach problems with the confidence and logic necessary to succeed.