Problem 25 - College Algebra

Question

The formula $$ \frac{W}{2}-3 H=53 $$ models the recommended weight \(W\), in pounds, for a male, where \(H\) represents the man's height, in inches, over 5 feet. What is the recommended weight for a man who is 6 feet, 3 inches tall?

Step-by-Step Solution

Answer

According to the given model, the recommended weight for a man who is 6 feet, 3 inches tall is 196 pounds.

Step 1: Convert the height into inches

First, you need to convert the height into inches. The man is 6 feet, 3 inches tall. As 1 foot is 12 inches, 6 feet convert into \(6 \times 12 = 72\) inches. Hence, the total height in inches is \(72 + 3 = 75\) inches. As the model considers height over 5 feet, we also have to subtract 5 feet in inches (which equals \(5 \times 12 = 60\) inches) from the total height. That leaves us with \(75 - 60 = 15\) inches. This will be our \(H\) value in the formula.

Step 2: Substitute the height (H) into the equation and solve for W

Substitute \(H = 15\) into the equation \(\frac{W}{2}-3 H=53\). We get \(\frac{W}{2} - 3 \times 15 = 53\). Simplify the right side of the equation to get \(\frac{W}{2} - 45 = 53\). To isolate \(\frac{W}{2}\), add 45 to both sides to get \(\frac{W}{2} = 53 + 45 = 98\). As the last step, multiply both sides of the equation by 2 to solve for \(W\). Thus, \(W = 98 \times 2 = 196\).

Step 3: Interpret the result

The solution \(W = 196\) represents the recommended weight in pounds for a man who is 6 feet, 3 inches tall according to the given model. This is the final answer.

Key Concepts

Linear Equations

Linear equations are foundations of algebra and represent relationships between two variables in the form of a straight line when plotted on a graph. They are characterized by their first-degree exponents, meaning the variables are not raised to any power higher than one.

The standard form of a linear equation is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants and \(x\) and \(y\) are the variables. Solving these equations usually involves finding the value of one variable when the other is known. In the exercise provided, we see a linear equation modeling weight as a function of height for an individual. The key steps involve isolating the variable you wish to solve for (in this case, \(W\), which stands for weight) and performing algebraic operations to find its value.

Unit Conversion

Unit conversion is essential in solving real-world problems because measurements can be represented in different units. For instance, to communicate effectively in various areas, such as science, construction, and everyday life, knowing how to convert measurements from one unit to another is crucial.

In our example, we see height initially given in feet and inches. The unit conversion involves transforming this mixed measurement into inches only. This is done through knowing the equivalent values, such as 1 foot equals 12 inches. Here's a useful tip: always write down the units when performing each step of the conversion to avoid confusion and ensure accuracy. Once you've converted to the appropriate units, you can then proceed with solving the equation as it aligns with the given formula.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. These expressions, unlike equations, do not have an equals sign. They are used to represent quantities and to set up equations that describe some statement or rule.

In the exercise above, the algebraic expression was \(\frac{W}{2}-3H\). The expression describes the relationship between a man's weight and height in the context of a recommended weight model. It's important to understand how to work with algebraic expressions as they form the basis for solving equations. Simplifying expressions by performing operations such as addition, subtraction, multiplication, or division prepares us for manipulating and eventually solving equations.