Problem 82 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

SURFACE AREA OF A HUMAN BODY The surface area \(S\) of the body of an average person 4 feet tall who weighs \(w \mathrm{lb}\) changes at the rate $$ S^{\prime}(w)=110 w^{-0.575} \quad \text { in }^{2} / \mathrm{lb} $$ The body of a particular child who is 4 feet tall and weighs \(50 \mathrm{lb}\) has surface area \(1,365 \mathrm{in}^{2}\). If the child gains \(3 \mathrm{lb}\) while remaining the same height, by how much will the surface area of the child's body increase?

Step-by-Step Solution

Answer

The surface area will increase by approximately 18.5571 \text{ in}^2.

Step 1: Identify the given information

The rate of change of surface area with respect to weight is given as \( S'(w) = 110 w^{-0.575} \). The initial weight of the child is 50 lb, and the child gains 3 lb.

Step 2: Calculate the weight after the gain

The new weight of the child after gaining 3 lb is \( w = 50 + 3 = 53 \) lb.

Step 3: Use the derivative to find the rate of change at the initial weight

Substitute \( w = 50 \) into the derivative: \[ S'(50) = 110 \times 50^{-0.575} \]

Step 4: Simplify the expression

Calculate \( 50^{-0.575} \): \[ 50^{-0.575} \approx 0.056234 \] Therefore, \[ S'(50) = 110 \times 0.056234 \approx 6.1857 \text{ in}^2/\text{lb} \]

Step 5: Determine the increase in surface area

Multiply the rate of change by the weight gain: \[ \text{Increase in surface area} = S'(50) \times 3 = 6.1857 \times 3 \approx 18.5571 \text{ in}^2 \]

Key Concepts

Rate of Change

In this exercise, we are dealing with the rate of change of the surface area of a human body with respect to weight. The concept of rate of change refers to how one quantity changes in response to another. Here, it tells us how the surface area changes when the weight changes. Mathematically, this is represented by a derivative. The rate of change given is \( S'(w) = 110 w^{-0.575} \).
This means for any weight \( w \), we can calculate how much the surface area will change per unit change in weight. For example, when the child's weight is 50 lb, \( S'(50) \) tells us how many square inches the surface area will increase (or decrease) per additional pound.

Surface Area Calculation

Calculating the surface area of a human body involves understanding how it changes with weight. From the given rate of change function, we can calculate the exact change in surface area when the weight changes. Initially, the child’s weight is 50 lb, and the rate of change of surface area at this weight is calculated by substituting \( w = 50 \) into the derivative \( S'(w) \).
This approach simplifies the calculation process and helps determine the additional or reduced surface area for any weight gain or loss.

Derivatives

Derivatives are a fundamental part of calculus. They represent the rate at which a function is changing at any given point. In this exercise, the derivative \( S'(w) = 110 w^{-0.575} \) models the rate of change of surface area with respect to weight. By evaluating this derivative at \( w = 50 \), we find the precise rate of change for a child weighing 50 lb, which is \( 6.1857 \text{ in}^2/\text{lb} \).
This value represents the increase in surface area per pound. Multiplying this rate by the weight gain (3 lb) gives the total increase in surface area.

Weight Gain Impact

Weight gain in a child impacts their physical characteristics, including the surface area of their body. Using calculus, we can precisely quantify this impact. In this scenario, a 3 lb weight gain results in an increase in surface area. By multiplying the rate of change \( 6.1857 \text{ in}^2/\text{lb} \) by the weight gain (3 lb), we conclude the surface area increases by \( 18.5571 \text{ in}^2 \).
This calculation shows a direct correlation between weight gain and the surface area increase, emphasizing how changes in one physical attribute can affect another.