We are asked to estimate the maximum error in the calculated surface area $S$ using the formula:

$S = 0.142 w^{0.314} h^{0.79}$

Given that the errors in the measurements of weight ($w$) and height ($h$) are at most 2%, we need to estimate the maximum relative error in the surface area.

Step 1: Differentiate the equation

The relative error in $S$ can be approximated using partial derivatives. Taking the natural logarithm of both sides of the equation $S = 0.142 w^{0.314} h^{0.79}$, we get:

$\ln(S) = \ln(0.142) + 0.314 \ln(w) + 0.79 \ln(h)$

Now, differentiating both sides with respect to $w$ and $h$, we obtain:

$\frac{dS}{S} = 0.314 \frac{dw}{w} + 0.79 \frac{dh}{h}$

Step 2: Maximum error approximation

Since the error in both $w$ and $h$ is given as 2%, the maximum errors in $\frac{dw}{w}$ and $\frac{dh}{h}$ are:

$\frac{dw}{w} = 0.02 \quad \text{and} \quad \frac{dh}{h} = 0.02$

Substitute these values into the equation for $\frac{dS}{S}$:

$\frac{dS}{S} = 0.314(0.02) + 0.79(0.02)$

Simplifying:

$\frac{dS}{S} = 0.00628 + 0.0158 = 0.02208$

Step 3: Express the result as a percentage

The relative error in $S$ is approximately $0.02208$, or 2.208%. Therefore, the maximum error in the calculated surface area is:

$\boxed{2.208\%}$

This means that the maximum error in the surface area $S$ is approximately 2.208%, given that the errors in both $w$ and $h$ are at most 2%.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How do you calculate the relative error in more complex models?
2. What happens if the error in $w$ and $h$ differs from each other?
3. How does the coefficient in the surface area formula affect the error calculation?
4. What is the significance of the exponents on $w$ and $h$ in determining sensitivity to errors?
5. Can we find a general formula for maximum error estimation for any power-law relationship?

Tip: When estimating maximum error, small percentage errors in input values lead to approximations in the overall output using differential methods like partial derivatives.