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:
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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 \( \Delta q = 0.01 \times 6.15 = 0.0615 , \text{cm} \).
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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 \( q \): \[ \text{Relative Error in } q = \frac{\Delta q}{q} = \frac{0.0615}{6.15} \approx 0.01 \text{ (or } 1%\text{)} \]
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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 \( q \).