Question

One of several empirical formulas that relates the surface area \(S\) of a human body to the height \(h\) and weight \(w\) of the body is the Mosteller formula \(S(h, w)=\frac{1}{60} \sqrt{h w},\) where \(h\) is measured in centimeters, \(w\) is measured in kilograms, and \(S\) is measured in square meters. Suppose that \(h\) and \(w\) are functions of \(t\). a. Find \(S^{\prime}(t)\). b. Show that the condition that the surface area remains constant as \(h\) and \(w\) change is \(w h^{\prime}(t)+h w^{\prime}(t)=0\). c. Show that part (b) implies that for constant surface area, \(h\) and \(w\) must be inversely related; that is, \(h=C / w,\) where \(C\) is a constant.

Short answer

Answer: When the surface area of a human body remains constant, the height (h) and weight (w) must be inversely related, with the relationship given by \(h = \frac{C}{w}\), where \(C\) is a constant.

Step-by-step solution

Differentiate the Mosteller formula with respect to time

Differentiate the Mosteller formula with respect to time (t) using the chain rule. \(\frac{dS}{dt} = \frac{d}{dt}(\frac{1}{60} \sqrt{h(t) w(t)})\) Since \(h\) and \(w\) are both functions of \(t\), we will need to use the chain rule when differentiating: \(\frac{dS}{dt} = \frac{1}{60} \frac{d}{dt}(\sqrt{h(t) w(t)}) = \frac{1}{60} \frac{1}{2\sqrt{h(t) w(t)}} \frac{d}{dt}(h(t)w(t))\) Now, using the product rule: \(\frac{dS}{dt} = \frac{1}{120\sqrt{h(t) w(t)}} (h(t)w'(t) + w(t)h'(t))\) So, we have found \(S^{\prime}(t)\): \(S^{\prime}(t) = \frac{1}{120\sqrt{h(t) w(t)}} (h(t)w'(t) + w(t)h'(t))\) #b. Show that the condition that the surface area remains constant as \(h\) and \(w\) change is \(w h^{\prime}(t) + h w^{\prime}(t) = 0\).

Set the rate of change of surface area to zero

To show that the surface area remains constant, the rate of change of surface area with respect to time should be zero: \(S^{\prime}(t) = 0\) Now, plug in the expression we found in part a and set it equal to zero: \(\frac{1}{120\sqrt{h(t) w(t)}} (h(t)w'(t) + w(t)h'(t))=0\)

Simplify the expression

If we multiply both sides of the equation by \(120\sqrt{h(t) w(t)}\), we get: \(h(t)w'(t) + w(t)h'(t) = 0\) #c. Show that part (b) implies that for constant surface area, \(h\) and \(w\) must be inversely related; that is, \(h=C / w\), where \(C\) is a constant.

Separate variables

We have the following relationship from part (b): \(h(t)w'(t) + w(t)h'(t) = 0\) Now we'll isolate both derivatives on the other side: \(h(t)w'(t) = -w(t)h'(t)\) Divide both sides by \(h(t)w(t)\): \(\frac{w'(t)}{w(t)} = -\frac{h'(t)}{h(t)}\)

Integrate both sides

Now integrate both sides with respect to \(t\): \(\int \frac{w'(t)}{w(t)}dt = \int -\frac{h'(t)}{h(t)} dt\) Using the properties of integrals, it can be written as: \(\ln |w(t)| = -\ln |h(t)| + C_1\) Where \(C_1\) is the constant of integration.

Solve for the relationship

Now, we'll rewrite the expression using exponential function: \(|w(t)| = \frac{C_2}{|h(t)|}\) Where \(C_2 = e^{C_1}\) is also a constant. If the surface area remains constant, the absolute values are not necessary as both height and weight must be positive. Thus, we can rewrite the equation as: \(h(t) = \frac{C}{w(t)}\) Where \(C = \frac{1}{C_2}\) is also a constant. In conclusion, if the surface area of a human body remains constant as height and weight change, \(h\) and \(w\) must be inversely related, with \(h = \frac{C}{w}\), where \(C\) is a constant.

Key concepts

Differentiation

Differentiation is a central concept in calculus, describing the process of finding the derivative of a function. The derivative represents how a function changes as its input changes. In this problem, we are interested in how the surface area of a human body, determined by height and weight, changes over time. To express this change mathematically, we compute the derivative of the Mosteller formula with respect to time, noted as \( S'(t) \). This derivative, \( S'(t) \), helps us understand how the surface area changes as both height, \( h(t) \), and weight, \( w(t) \), vary with time \( t \). Differentiating complex expressions often requires rules that simplify the process, like the chain rule and the product rule.

Chain Rule

The chain rule is a fundamental differentiation technique used for finding derivatives of composite functions. In essence, it helps distinguish how the inner and outer functions in a composite equation contribute to the overall derivative. For example, the Mosteller formula involves \( \sqrt{h(t)w(t)} \). Since both \( h(t) \) and \( w(t) \) are functions of \( t \), the chain rule helps differentiate \( \sqrt{h(t)w(t)} \) effectively.
To apply the chain rule here, we first differentiate the outer layer function, which is the square root, leaving the inner function, \( h(t)w(t) \), untouched. Then, we multiply that result by the derivative of the inner function, \( h(t)w(t) \). This step is crucial in obtaining the overall derivative \( S'(t) \), contributing to the understanding of how both height and weight affect the surface area over time.

Product Rule

The product rule is an indispensable differentiation rule when dealing with functions that are products of two or more functions. In our situation, we encounter \( h(t)w(t) \), a product of the height and weight functions, each varying with time. The product rule states that the derivative of a product of two functions \( u(t) \) and \( v(t) \) is \( u'(t)v(t) + u(t)v'(t) \).
Applying this to \( h(t)w(t) \), we obtain \( h(t)w'(t) + w(t)h'(t) \). This expression reflects how the product of these two functions changes over time. In our problem, it integrates with the chain rule outcome to fully determine the time derivative of the surface area. Using both rules effectively allows calculus students to confidently tackle expressions involving multiple functions.

Inverse Relationships

In the context of differentiation, inverse relationships refer to situations where one variable is inversely proportional to another. From part (b) of the problem, if we want the surface area to remain constant, the derivative \( S'(t) \) must equal zero. This condition leads us to find that \( h(t)w'(t) + w(t)h'(t) = 0 \), indicating that changes in height and weight must counterbalance each other.
Solving this equation reveals an interesting inverse relationship between \( h(t) \) and \( w(t) \), suggesting that \( h = C/w \), where \( C \) is a constant, if the surface area remains unchanged. In essence, if height increases, weight must decrease proportionally to maintain a constant surface area, reflecting their inverse proportionality.