To estimate the maximum error in the calculated surface area $S$, we need to apply the concept of differentials and relative errors.

Given: $S = 0.142w^{0.314}h^{0.79}$ Where:

• $w$ is the weight in pounds
• $h$ is the height in inches
• $S$ is the surface area in square feet

To find the maximum relative error in $S$, we'll differentiate the formula for $S$ with respect to both $w$ and $h$. The general approach is to use the following formula for the differential:

$\frac{dS}{S} \approx \left| \frac{\partial S}{\partial w} \cdot \frac{dw}{S} \right| + \left| \frac{\partial S}{\partial h} \cdot \frac{dh}{S} \right|$

Step 1: Differentiate $S$ with respect to $w$ and $h$

The partial derivatives of $S$ are:

$\frac{\partial S}{\partial w} = 0.142 \cdot 0.314 \cdot w^{0.314 - 1} \cdot h^{0.79} = 0.044588 \cdot w^{-0.686} \cdot h^{0.79}$

$\frac{\partial S}{\partial h} = 0.142 \cdot 0.79 \cdot w^{0.314} \cdot h^{0.79 - 1} = 0.11218 \cdot w^{0.314} \cdot h^{-0.21}$

Step 2: Compute the relative error in $S$

We know that the maximum relative errors in $w$ and $h$ are 2%, or 0.02 in decimal form. So, $\frac{dw}{w} = \frac{dh}{h} = 0.02$.

Now, the differential $\frac{dS}{S}$ becomes:

$\frac{dS}{S} \approx \left| 0.314 \cdot \frac{dw}{w} \right| + \left| 0.79 \cdot \frac{dh}{h} \right|$

Substituting $\frac{dw}{w} = 0.02$ and $\frac{dh}{h} = 0.02$:

$\frac{dS}{S} \approx 0.314 \cdot 0.02 + 0.79 \cdot 0.02 = 0.00628 + 0.0158 = 0.02208$

Step 3: Convert relative error into percentage

The maximum relative error in $S$ is $0.02208$, or approximately 2.21%. Thus, the maximum error in the calculated surface area is 2.21%.

Final Answer:

The estimate of the maximum error in $S$ is 2.21%.

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Do you want more details or have any questions?

Here are 5 related questions:

1. How is the formula for relative error derived in differentials?
2. What happens if the relative errors in $w$ and $h$ are different?
3. How would you calculate the error in volume for a similar problem?
4. How does the exponent affect the sensitivity to error in this case?
5. Can we calculate the absolute error instead of relative error?

Tip: When using differentials to estimate error, the relative error in a function depends on the exponents and coefficients of the variables in the equation.