Problem 45 - Algebra A Combined Function

Question

Solve each equation for the indicated variable. $$ B=\frac{705 w}{h^{2}} \text { for } w \text { (Health: body-mass index) } $$

Step-by-Step Solution

Answer

The solution for \( w \) is \( w = \frac{B \cdot h^2}{705} \).

Step 1: Understand the Equation

The given equation is \( B = \frac{705 w}{h^2} \). This equation represents the body-mass index equation where \( B \) is the body-mass index, \( w \) is the weight, and \( h \) is the height. Our goal is to solve for \( w \).

Step 2: Isolate \( w \)

To isolate \( w \) in the equation \( B = \frac{705 w}{h^2} \), we need to get \( w \) by itself on one side of the equation. We start by multiplying both sides by \( h^2 \) to get rid of the fraction: \( B \cdot h^2 = 705 w \).

Step 3: Solve for \( w \)

Now, we isolate \( w \) by dividing both sides of the equation by 705: \( w = \frac{B \cdot h^2}{705} \). Now we have \( w \) expressed in terms of \( B \) and \( h \).

Key Concepts

Body-Mass Index

The Body-Mass Index (BMI) is a simple, yet classic way to measure a person's body weight relative to their height. This computation helps in assessing whether an individual falls within a particular weight category based on predefined criteria. The formula for calculating BMI is given by:

• \( B = \frac{705w}{h^2} \)

Where,

• \( B \) represents the Body-Mass Index
• \( w \) stands for weight in pounds
• \( h \) is height in inches

This formulation helps categorize a person into underweight, normal weight, overweight, or obese categories. It's important to note that BMI is a screening tool and does not diagnose any health conditions. However, it provides a quick insight into potential unhealthy weight levels.

Variable Isolation

Variable isolation is an essential algebraic concept where the aim is to focus on one specific variable in an equation. When solving an equation, it's important to "isolate" the variable that you're solving for. In this context, if you're looking to solve for the weight \( w \) in the BMI equation, you aim to get \( w \) on its own on one side of the equation.This typically involves reversing operations applied to the variable. Using our given equation, \( B = \frac{705w}{h^2} \), to isolate \( w \), you will:

• Multiply both sides by \( h^2 \) to eliminate the fraction, leading to \( B \cdot h^2 = 705w \).
• The last step is to divide by 705 to completely isolate \( w \), resulting in \( w = \frac{B \cdot h^2}{705} \).

Each step in the isolation process helps inch closer to unveiling the desired variable, making it easier to solve the problem at hand.

Algebraic Manipulation

Algebraic manipulation is a technique used to rearrange equations and expressions to solve for unknown variables. It involves using basic algebraic operations such as addition, subtraction, multiplication, and division to simplify or solve equations. In the context of our exercise, the equation \( B = \frac{705w}{h^2} \) needs manipulation to solve for \( w \).The logic behind these manipulations is as follows:

• Get rid of fractions by multiplying through by the denominator.
• Adjust each side of the equation through division to isolate the variable.

The equation transformation starts with removing the division presented in the fraction by multiplying each side by \( h^2 \). Then, by dividing both sides by 705, we achieve a formula solely in terms of \( w \). This method of manipulation not only solves the equation but also demonstrates the profound power of algebraic thinking in simplifying complex formulas.