https://file.aiquickdraw.com/mathbot/file/1725500240534syfeox5e.jpg

The image shows the Mosteller formula for approximating the surface area \( S \) in \( m^2 \) of a human, which is given by:

\[
S = \frac{\sqrt{hw}}{60}
\]

where \( h \) is the person’s height in cm and \( w \) is the person’s weight in kg. The problem asks to compute the partial derivative \( \frac{\partial S}{\partial h} \), which is the rate of change of surface area with respect to height \( h \), keeping weight \( w \) constant.

### Step-by-Step Solution:
1. **Start with the formula:**

\[
S = \frac{\sqrt{hw}}{60}
\]

2. **Differentiate \( S \) with respect to \( h \):**
   Since \( w \) is treated as constant in this partial derivative, use the chain rule. The derivative of \( \sqrt{hw} \) with respect to \( h \) is:

\[
\frac{d}{dh} \left( \sqrt{hw} \right) = \frac{1}{2\sqrt{hw}} \cdot w
\]

3. **Now differentiate \( S \):**

\[
\frac{\partial S}{\partial h} = \frac{1}{60} \cdot \frac{1}{2\sqrt{hw}} \cdot w = \frac{w}{120 \sqrt{hw}}
\]

Thus, the partial derivative \( \frac{\partial S}{\partial h} \) is:

\[
\frac{\partial S}{\partial h} = \frac{w}{120 \sqrt{hw}}
\]

Let me know if you'd like further clarification or have any questions!

### Expanding on this:
1. What does the partial derivative tell us about the relationship between surface area and height?
2. How would the derivative change if both height and weight are allowed to vary?
3. What is the geometric interpretation of this derivative in terms of the surface area of the body?
4. Can we compute the partial derivative with respect to \( w \) in a similar way?
5. How could this model be used to estimate surface area in clinical settings?

### Tip:
When computing partial derivatives, always treat the variable you are differentiating with respect to as the only variable that changes, holding the others constant.

The Mosteller formula for approximating the surface area, S, in m² of a human, is given by S = √(hw)/60, where h is the person's height in cm and w is the person's weight in kg. The problem asks to compute the partial derivative dS/dh.

This page explains how to compute the partial derivative of the Mosteller formula for approximating human surface area: S = √(hw) / 60. Learn how the chain rule applies and find the rate of change of surface area with respect to height.

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