Problem 60 - Algebra A Combined Function

Question

Calculating body-mass index \((B M I)\) is a way to gauge whether a person should lose weight. Doctors recommend that body-mass index values fall between 19 and \(25 .\) The formula for body-mass index \(B\) is \(B=\frac{705 w}{h^{2}},\) where \(w\) is weight in pounds and \(h\) is height in inches. A doctor recorded a body-mass index of 47 on a patient's chart. Later, a nurse notices that the doctor recorded the patient's weight as 240 pounds but neglected to record the patient's height. Explain how the nurse can use the information from the chart to find the patient's height. Then find the height.

Step-by-Step Solution

Answer

The patient's height is 60 inches.

Step 1: Understanding the BMI Formula

The formula for the body-mass index (BMI) is given by \( B = \frac{705w}{h^2} \), where \( B \) is the BMI, \( w \) is the weight in pounds, and \( h \) is the height in inches. We are given \( B = 47 \) and \( w = 240 \) pounds and need to find \( h \).

Step 2: Rearranging the Formula for Height

To find \( h \), we rearrange the BMI formula. Starting from \( B = \frac{705w}{h^2} \), we multiply both sides by \( h^2 \) to get \( Bh^2 = 705w \). Then, divide both sides by \( B \) to find \( h^2 = \frac{705w}{B} \).

Step 3: Substituting Known Values

Now, substitute the known values \( w = 240 \) pounds and \( B = 47 \) into the equation \( h^2 = \frac{705 \times 240}{47} \).

Step 4: Calculating the Right Side of the Equation

Calculate the numerator: \( 705 \times 240 = 169200 \). Then, divide by 47 to find \( h^2 = \frac{169200}{47} \). Performing the division gives \( h^2 = 3600 \).

Step 5: Solving for Height \(h\)

To find \( h \), take the square root of both sides: \( h = \sqrt{3600} \). The square root of 3600 is 60.

Step 6: Verifying the Result

Verify the calculation by plugging \( h = 60 \) back into the original BMI formula: \( B = \frac{705 \times 240}{60^2} = \frac{169200}{3600} = 47 \). The BMI calculation verifies as correct.

Key Concepts

Algebraic Manipulation

Understanding algebraic manipulation is key when working with equations to isolate a desired variable, such as finding the height in a BMI problem. Algebraic manipulation means rearranging equations to find an unknown value. This skill is essential in numerous real-life applications, like calculating BMI, a person's credit score, or even distances in physics.

In the BMI calculation problem, we start with the formula \( B = \frac{705w}{h^2} \). The goal is to isolate \( h \) so we can find a patient's height. To do this, we follow these steps:

• First, multiply both sides by \( h^2 \) to eliminate the fraction: \( Bh^2 = 705w \)
• Next, divide both sides by \( B \) to isolate \( h^2 \): \( h^2 = \frac{705w}{B} \)

These steps demonstrate how algebraic manipulation helps transform the original equation into a form where the desired variable is by itself. This step-by-step change is crucial for solving the problem accurately.

Mathematical Formulas

Mathematical formulas are predefined equations that describe relationships between different variables. They allow us to compute one variable based on the values of others. The BMI formula is a perfect example of using mathematical formulas in health and fitness.

The formula \( B = \frac{705w}{h^2} \) connects a person's body mass index (BMI) to their weight and height. This equation forms the basis to calculate BMI given weight and height, which are typically recorded at medical check-ups. By substituting known values into the formula, practitioners can monitor and advise patients about their health.

Using mathematical formulas can seem daunting at first, but they simply represent the relationships you work with in the problem. Think of formulas as tools in your toolkit. The more comfortable you become using them, the easier it gets to solve related problems.

Problem Solving

Problem solving in mathematics involves understanding a problem, devising a plan, carrying out that plan, and evaluating the results. Start by identifying what you know and what you need to find out. In the BMI exercise, you know the BMI and weight, but not the height.

The first step is understanding the given BMI formula. Next, use algebraic manipulation to rearrange the formula to solve for height. Then, replace the known values into the reformulated equation. Calculate carefully and take the square root to find the height.

Finally, verify your calculations to ensure the solution is correct. Good problem solving not only involves performing calculations, but also making sure that those calculations align with the requirements of the problem. By verifying your answer, as done by recalculating BMI with the found height, you confirm the solution is logical and accurate.

Effective problem solving demands patience and practice. By following these structured steps, it becomes easier to tackle and solve more complex problems.