Problem 48 - College Algebra

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 \(\mathrm{kg}\) and \(h=\) height in \(\mathrm{cm} .\) Find the height of a 72 -kg female to the nearest cm whose \(B S A=1.8 .\)

Step-by-Step Solution

Answer

The height is 162 cm.

Step 1: Understand the Formula

The given formula for BSA is \( \text{BSA} = \sqrt{\frac{wh}{3600}} \), where \( w \) is the weight in kilograms and \( h \) is the height in centimeters. We are asked to find the height \( h \) when \( \text{BSA} = 1.8 \) and \( w = 72 \text{ kg} \).

Step 2: Substitute the Known Values

Replace the given values in the formula. So, the formula becomes \( 1.8 = \sqrt{\frac{72 \times h}{3600}} \).

Step 3: Square Both Sides

To eliminate the square root, square both sides of the equation: \( 1.8^2 = \frac{72 \times h}{3600} \). This simplifies to \( 3.24 = \frac{72 \times h}{3600} \).

Step 4: Solve for Height, h

Multiply both sides by 3600 to isolate \( 72h \): \( 3.24 \times 3600 = 72h \). This gives us \( 11664 = 72h \).

Step 5: Calculate the Value of h

Divide both sides by 72 to solve for \( h \): \( h = \frac{11664}{72} \). This yields \( h = 162 \).

Step 6: Round to Nearest Integer

The problem asks for height in the nearest centimeter. Since the calculated \( h \) is 162, we don't need to round further.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is like solving a puzzle using math rules. When given an equation, you can rearrange the pieces to isolate a particular variable, such as solving for height in our body surface area (BSA) formula. To manipulate an equation, you'll need to perform operations like addition, subtraction, multiplication, or division on both sides. This keeps the equation balanced, just like a scale.
For example, in our scenario, we wanted to solve for height (\( h \)) and began by simplifying the equation through substitution and other operations. By squaring both sides of the equation, multiplying, and then dividing, we cleverly isolated \( h \) to find the solution. Remember, the key is to always perform the same operation on each side of the equal sign!

Square Roots

Taking the square root of a number is essentially asking "what number, when multiplied by itself, equals that original number?" In the BSA formula, the square root is used to calculate an area from a ratio of weight to height. When solving our exercise, we dealt with the square root term \( \sqrt{\frac{wh}{3600}} \). By squaring both sides, you eliminate the square root and simplify the process of solving for \( h \).
This is a handy technique because it allows us to use basic arithmetic on more complex expressions. Just remember, squaring a number and taking the square root are inverse operations—like pressing undo! Using this method is a great way to transform an otherwise complicated equation into something more manageable.

Substitution Method

The substitution method involves replacing a variable with a given value to simplify an equation. Think of it like filling in the blanks with what you already know. In our BSA problem, we substituted the known values of weight (\( w = 72 \text{ kg} \)) and BSA (\( 1.8 \)) into the equation to solve for height (\( h \)).
So when we substituted these values, the equation \( 1.8 = \sqrt{\frac{72 \times h}{3600}} \) was formed. This made it easier to work with and apply further steps like squaring and solving for \( h \). Fundamentally, substitution helps streamline the problem-solving process by reducing the number of unknowns, thus bringing the solution clearly into view.

Solving Equations

Solving equations is like unwrapping a present; you're looking to reveal the unknowns beneath. In our problem, solving for \( h \) involved a systematic approach of applying algebraic methods and logical steps. The ultimate goal was to isolate \( h \) and find its value.

• First, we substituted the known values into the equation.
• Next, we squared both sides to remove the square root.
• Then, we multiplied across to isolate the term involving \( h \).
• Finally, we divided to solve for \( h \).

This step-by-step unraveling is crucial, as it breaks down the process into digestible tasks. Solving equations doesn't need to be daunting; when broken into parts, it becomes a straightforward journey to uncover the unknown.