Question

The body surface area (BSA) of a 180-centimeter-tall (about six-feet-tall) person is modeled by $$ B=0.1 \sqrt{5 w} $$ where \(B\) is the BSA (in square meters) and \(w\) is the weight (in kilograms). Use differentials to approximate the change in the person's BSA when the person's weight changes from 90 kilograms to 95 kilograms.

Short answer

The change in the person's body surface area (BSA) when their weight changes from 90 kg to 95 kg is approximately 0.075 square meters.

Step-by-step solution

Equation Differentiation

First, differentiate the provided equation of body surface area \(B\), given as \(B = 0.1 \sqrt{5w}\). By using chain rule for differentiation, we would get \(dB/dw = 0.1/2 \sqrt{5/w}\) or \(dB/dw = 0.05 \sqrt{5/w}\).

Differentials Calculation

Next, calculate the differential \(dB\) when weight changes by \(dw = 95 - 90 = 5\) kg. At \(w = 90\) kg, the differential is \(dB = (dB/dw) \cdot dw = 0.05 \sqrt{5/90} \cdot 5\).

Value Substitution

Substitute the given values into the equation for\(dB\). So, \(dB = 0.05 \sqrt{5/90} \cdot 5 \approx 0.075\) square meters.

Key concepts

Body Surface Area

Understanding body surface area (BSA) is crucial, especially in medical fields. BSA is a measurement of the total surface of the human body. It's often used in dosages for medications, as it provides a more accurate measurement than using weight alone. The formula provided, \( B = 0.1 \sqrt{5w} \), models BSA based on weight \( w \). Here, \( B \) is the body surface area in square meters, and \( w \) is weight in kilograms.

This calculation shows how BSA scales with changes in weight by incorporating a square root function, which reflects the non-linear relationship between weight and surface area. In this context, changes in weight correspond to changes in BSA, allowing us to approximate these changes mathematically.

Chain Rule

The chain rule is a fundamental concept in differential calculus. It's applied when differentiating composite functions—functions within functions. In this exercise, the chain rule helps us differentiate \( B = 0.1 \sqrt{5w} \).

To understand the chain rule's role, consider \( f(g(x)) \), where \( f \) and \( g \) are functions. The derivative is \( f'(g(x)) \cdot g'(x) \).

• The outer function here is \( 0.1 \sqrt{} \), and it's composed with the inner function \( 5w \).
• First, differentiate the outer function as if the inner one were a simple variable.
• Then, multiply by the derivative of the inner function \( 5w \).

This yields \( dB/dw = 0.05 \sqrt{5/w} \), allowing us to understand how BSA changes as weight varies.

Differential Approximation

Differential approximation is a technique to estimate how a small change in one variable affects another. This is particularly useful in cases where direct calculations may be complex. It provides a linear approximation of how a function changes using its derivative.

In the problem, the goal is to approximate how the BSA changes as weight increases from 90 kg to 95 kg. We use the differential \( dB \) to find this rate of change. Here's a breakdown of the steps:

• Calculate the differential change in weight, \( dw = 95 - 90 = 5 \) kg.
• Using \( dB = (dB/dw) \cdot dw \), substitute the derivative \( dB/dw \) calculated earlier.
• Plug in the initial weight \( w = 90 \) kg into the derivative expression.

The result is an approximate change \( dB \approx 0.075 \) square meters, showing us the sensitivity of BSA to weight changes.