BMI=W/H^2 if the errors in weight W and height H are both 1% what is percentage error of BMI

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To solve for the percentage error in BMI, we need to consider how errors propagate through a formula.

The formula for Body Mass Index (BMI) is given by:

\[
BMI = \frac{W}{H^2}
\]

Where:
- \(W\) is the weight,
- \(H\) is the height.

Now, we are told that the errors in weight (\(W\)) and height (\(H\)) are both 1%. We need to find the percentage error in BMI. 

### Step 1: Error propagation rule
For any function of multiple variables, the relative (percentage) error in the function can be approximated using the following rule for functions with independent variables:

\[
\frac{\Delta f}{f} \approx \left| n \right|\frac{\Delta x_1}{x_1} + \left| m \right|\frac{\Delta x_2}{x_2}
\]

Where \(f = x_1^n x_2^m\) and \(\frac{\Delta f}{f}\) is the relative error in \(f\).

In our case:
- \(f = \frac{W}{H^2}\), so we have:
  - \(W\) raised to the power 1 (\(n = 1\)),
  - \(H\) raised to the power \(-2\) (\(m = -2\)).

### Step 2: Apply the error propagation formula
The relative error in BMI can be written as:

\[
\frac{\Delta BMI}{BMI} \approx \left| 1 \right| \cdot \frac{\Delta W}{W} + \left| -2 \right| \cdot \frac{\Delta H}{H}
\]

Since the percentage error in both weight (\(W\)) and height (\(H\)) is 1%, we substitute the errors:

\[
\frac{\Delta BMI}{BMI} = 1 \times 1\% + 2 \times 1\%
\]

\[
\frac{\Delta BMI}{BMI} = 1\% + 2\%
\]

\[
\frac{\Delta BMI}{BMI} = 3\%
\]

Thus, the percentage error in BMI is **3%**.

Would you like more details, or have any questions? Here are some follow-up questions for further exploration:

1. How would the percentage error change if the error in height were 2% instead of 1%?
2. What happens to the BMI error if the height error is reduced to zero?
3. If the weight error is doubled, how does it impact the total percentage error in BMI?
4. Can you apply the same error propagation method to a different formula, like kinetic energy?
5. How would correlated errors (errors not being independent) affect the error propagation?

**Tip:** Remember that errors in quantities that appear with higher powers in formulas contribute more to the overall percentage error!

BMI=W/H^2 if the errors in weight W and height H are both 1% what is percentage error of BMI

Learn how to calculate the percentage error in BMI when both weight and height have a 1% error. This solution uses the error propagation formula, showing that the total percentage error in BMI is 3%. Suitable for Grades 10-12 students studying error analysis.

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