Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, use a model for body surface area, BSA, such that \(B S A=\sqrt{\frac{w h}{3600}}\), where w= weight in kg and \(h=\) height in cm. Find the height of a \(72-\mathrm{kg}\) female to the nearest \(\mathrm{cm}\) whos \(B S A=1.8\).

Step-by-Step Solution

Answer

The height is 162 cm.

Step 1: Identify Given Values and Equation

We know the body surface area (BSA) formula is \( BSA = \sqrt{\frac{wh}{3600}} \). From the problem, \( BSA = 1.8 \) and \( w = 72 \) kg. We need to find the height \( h \).

Step 2: Substitute Values into the Formula

Substitute the given BSA and weight into the formula: \( 1.8 = \sqrt{\frac{72h}{3600}} \).

Step 3: Solve for Height Variable \( h \)

First, square both sides to eliminate the square root: \( 1.8^2 = \frac{72h}{3600} \). Calculate \( 1.8^2 = 3.24 \).

Step 4: Rearrange and Solve the Equation

Rearrange to \( 3.24 = \frac{72h}{3600} \) and multiply both sides by 3600 to get \( 11664 = 72h \).

Step 5: Isolate \( h \) and Calculate

Divide both sides by 72 to find \( h \): \( h = \frac{11664}{72} = 162 \).

Step 6: Round to Nearest cm

Since height needs to be to the nearest centimeter, \( h = 162 \). Considering the calculations were exact, we confirm \( h = 162 \) cm.

Key Concepts

Algebraic Formulas

Algebraic formulas are essential tools in solving mathematical equations. In this case, we used the body surface area (BSA) formula: \[ BSA = \sqrt{\frac{wh}{3600}} \] Here, \(w\) represents weight in kilograms and \(h\) represents height in centimeters. These formulas are used to describe relationships between different quantities. They allow us to rearrange and substitute values to isolate the desired variable.

• Start by identifying known values, such as \(BSA\) and \(w\).
• Next, substitute these values into the formula, which helps set up the equation to be solved.

Understanding algebraic formulas involves recognizing how to manipulate these equations to find an unknown value, such as height in this problem. This involves steps like squaring both sides to eliminate square roots or rearranging terms to isolate a variable.

Weight and Height Calculations

When calculating weight and height in mathematical problems, it is crucial to convert them into the appropriate units. In this exercise, weight is given in kilograms, and height is in centimeters, which is typical when calculating body surface area.

• Always ensure that units are consistent with those in the formula to avoid errors.
• Weight \(w\) and height \(h\) values are plugged into the equations using these units.

Using the formula \(BSA = \sqrt{\frac{wh}{3600}}\), ensures that calculations are correctly applied with the given measurements. By substituting the weight of 72 kg and solving for height \(h\), we found that \(h = 162\) cm.

Mathematical Problem Solving

Mathematical problem solving is a systematic approach that relies on logical reasoning to find a solution. Follow these general steps to tackle any problem efficiently:

• Identify what you need to find and the information provided.
• Choose the appropriate formula or concept related to your task, such as the BSA formula here.
• Carefully substitute known values, then manipulate the formula to isolate the unknown.
• Use arithmetic operations to solve step by step, verifying at each stage.

For example, starting with the given formula \(BSA = \sqrt{\frac{wh}{3600}}\), substituting \(BSA = 1.8\) and \(w = 72\), and solving for \(h\) involves these logical steps. Consistent practice in applying these steps develops proficiency in mathematical problem solving.