Question

Given that p=2.5 cm and q=6.15cm, find the relative error in pq. Explain in a simple that i can understand.

Answer 1

To find the relative error in the product \( pq \), we first need to understand the concept of relative error and how it applies when multiplying two measurements.

### Steps to Find Relative Error in the Product:

1. **Understand Relative Error for Each Measurement:**
- The relative error of a measurement is the absolute error divided by the actual value of the measurement.
- We assume here that the errors in \( p \) and \( q \) (let's denote them as \( \Delta p \) and \( \Delta q \)) are given. If they are not provided, we can assume a percentage error based on common practice. Let’s assume a 1% error for each:
- For \( p = 2.5 \, \text{cm} \), the absolute error \( \Delta p = 0.01 \times 2.5 = 0.025 \, \text{cm} \).
- For \( q = 6.15 \, \text{cm} \), the absolute error \( \Delta q = 0.01 \times 6.15 = 0.0615 \, \text{cm} \).

2. **Calculate the Relative Errors:**
- The relative error in \( p \):
\[
\text{Relative Error in } p = \frac{\Delta p}{p} = \frac{0.025}{2.5} = 0.01 \text{ (or } 1\%\text{)}
\]
- The relative error in \( q \):
\[
\text{Relative Error in } q = \frac{\Delta q}{q} = \frac{0.0615}{6.15} \approx 0.01 \text{ (or } 1\%\text{)}
\]

3. **Find the Relative Error of the Product \( pq \):**
When multiplying two measurements, the relative errors add up. So, the relative error in the product \( pq \) is given by:
\[
\text{Relative Error in } pq = \text{Relative Error in } p + \text{Relative Error in } q
\]
Therefore:
\[
\text{Relative Error in } pq = 0.01 + 0.01 = 0.02 \text{ (or } 2\%\text{)}
\]

### Conclusion:
The relative error in the product \( pq \) is thus approximately **2%**. This means that if you multiply \( p \) and \( q \), the result can be off by about 2% of its value due to the uncertainties in \( p \) and \( q \).