Problem 59 - Introductory Algebra for College Students

Question

An HMO pamphlet contains the following recommended weight for women: "Give yourself 100 pounds for the first 5 feet plus 5 pounds for every inch over 5 feet tall." Using this description, which height corresponds to an ideal weight of 135 pounds?

Step-by-Step Solution

Answer

The height corresponding to the ideal weight of 135 pounds is 67 inches.

Step 1: Understand the Relationship

According to the HMO pamphlet, the ideal weight for women is determined like this: 100 pounds for the first 5 feet, plus 5 pounds for every inch over 5 feet. If a woman is exactly 5 feet tall, her ideal weight would be 100 pounds. For every additional inch of height, her ideal weight increases by 5 pounds.

Step 2: Establish an Equation

The next step is to turn this relationship into an algebraic equation that represents ideal weight as a function of height. Let's denote height in inches as 'h' (with 5 feet corresponding to 60 inches), and the ideal weight as 'w'. The formula then becomes: \(w = 100 + 5*(h - 60)\). Since we know that the ideal weight W is 135 pounds, we can plug it into the equation: \(135 = 100 + 5*(h - 60)\).

Step 3: Solve for Height

Now, we just need to solve this equation to find the height. First, let's subtract 100 from both sides to get: \(35 = 5*(h - 60)\). Then divide both sides of the equation by 5 to get: \(7 = h - 60\). Adding 60 to both sides of the equation gives the solution: \(h = 67\). So the height that corresponds to an ideal weight of 135 pounds is 67 inches.

Key Concepts

Understand the Problem-Solving Steps

When tackling algebraic equations, following a clear problem-solving pathway can make the process straightforward and effective.
To understand any problem, it's important to:

• Study the problem statement: Determine what information is given and what is required.
• Identify the relationship: Recognize any patterns or rules that connect different elements of the problem.
• Translate to an equation: Converting relationships into algebraic form allows for easier manipulation and solution.
• Solve the equation: Execute calculations to find the unknown variable.
• Check the solution: Ensure the answer makes sense and satisfies the original problem.

Let's take the example provided: The task is to find the height given a weight of 135 pounds.
By following these steps, you can systematically solve the equation derived from the given relationship that weight increases by 5 pounds per inch over 5 feet.

Decoding Linear Equations

Linear equations form a significant foundation of algebra and are invaluable in expressing relationships between variables clearly.
A linear equation is typically in the form of:

• An expression that equals a constant, like: \(ax + b = c\).
• Variables raised only to the first power.

In our case, the equation \(135 = 100 + 5*(h - 60)\) captures the linear relationship between height in inches and weight.
To break it down:

• The term \(5*(h - 60)\) indicates a constant rate of change (5 pounds per inch above 5 feet).
• Subtracting and dividing during the solution process is essential, allowing isolation of the variable \(h\).

Solving this linear equation involves simple arithmetic operations to determine that a height of 67 inches corresponds to a weight of 135 pounds.

Applying Mathematical Modeling

Mathematical modeling involves using mathematical language and tools to represent real-world scenarios.
It simplifies complex situations using assumptions and approximations, turning them into solvable equations.

• In this example, the weight of a woman is modeled based on height using a straightforward linear equation.
• Assumptions include: The weight starts at 100 pounds for 5 feet and increases uniformly.

The model here describes:

• The given formula calculates an ideal weight, offering a reliable way to convert height into weight, enabling predictions or calculations.
• The utility of the model rests in its ability to be adapted to similar real-life calculations.

This approach highlights the power of mathematical modeling in providing clarity and decision-making tools for everyday problems like determining ideal weight at various heights.