Question

Show that under the assumption of small percentage tolerances there is a simple formula for the approximate percentage tolerance of the product of two intervals in terms of the tolerances of the factors. You may simplify the problem by assuming that all numbers are positive.

Short answer

The percentage tolerance of the product is approximately the sum of the tolerances of the factors.

Step-by-step solution

Understand the Problem

We need to find a formula for the percentage tolerance of a product of two intervals, given the tolerances of these intervals, assuming that the numbers are positive and tolerances are small.

Define the Variables and Assumptions

Let's say we have two numbers: \( a \) with a tolerance of \( t_a \)% and \( b \) with a tolerance of \( t_b \)%. The product of these numbers is \( ab \). We assume the tolerances are small, meaning \( t_a, t_b \ll 1 \).

Expand Using Percentage Tolerances

The value of the product with tolerances can be written as \((a + \, \Delta a)(b + \, \Delta b)\), where \( \Delta a \) is the tolerance of \( a \) and \( \Delta b \) is the tolerance of \( b \), with \(\Delta a = a \cdot \frac{t_a}{100} \) and \(\Delta b = b \cdot \frac{t_b}{100} \).

Expand the Product

The expansion leads to: \[ (a + \Delta a)(b + \Delta b) = ab + a \Delta b + b \Delta a + \Delta a \Delta b. \] Under our small tolerance assumption, the term \( \Delta a \Delta b \) is negligible.

Simplify the Expression

Omitting \( \Delta a \Delta b \), the expression becomes:\[ ab + a \Delta b + b \Delta a. \] Expressing the change in the product, \( \Delta(ab) \), gives us\[ \Delta(ab) = ab + a \Delta b + b \Delta a - ab = a \Delta b + b \Delta a. \]

Determine the Approximate Tolerance

Divide the tolerance change \( \Delta(ab) \) by the product \( ab \) and express in percentage:\[ \frac{\Delta(ab)}{ab} = \frac{a \Delta b + b \Delta a}{ab} = \frac{a b \cdot \frac{t_b}{100} + a b \cdot \frac{t_a}{100}}{a b} \approx \frac{t_b}{100} + \frac{t_a}{100}. \]Converting this back to percentage:\[ t_{ab} \approx t_a + t_b. \]

Key concepts

Percentage Tolerance

In mathematics, percentage tolerance is a way to express how much a given measurement can vary from its specified value. It provides a percentage that tells you how much a measurement can go above or below a specified range without being considered out of specification. For example, if a resistor is said to have a value of 100 Ohms with a tolerance of 5%, it means the actual resistance could be anywhere between 95 Ohms and 105 Ohms. Percentage tolerance is useful in various fields, especially in engineering and manufacturing, where precision is critical. It allows for a flexible but controlled variation in the production process. This ensures products function correctly within an acceptable range, rather than requiring exact perfection which is often impractical.

• It's calculated as the difference between the maximum and minimum permissible values, divided by the standard value, then multiplied by 100 to get a percentage.
• Being aware of this concept helps in understanding the reliability and performance of products and systems.

Tolerance in Products

When dealing with products of numbers, especially in contexts like engineering and physics, understanding their tolerance is crucial. Tolerance in products refers to the allowable percentage deviation in the result of multiplied components, such as dimensions in a mechanical part. In our exercise, we were asked to find the tolerance of a product of two numbers with known tolerances. If we have two values, each with a specified tolerance, then intuitively, the tolerance of their product depends on the tolerances of the individual factors. When calculating the tolerance of a product:

• The process involves expanding the product considering possible variations, and simplifying using the assumption that any interaction terms involving tolerances are negligible due to their small size.
• This simplified approach often assumes that errors in the factors do not compound dramatically unless the tolerances are very large.

Understanding tolerance in products helps ensure that the final outputs of calculations or manufactured products remain within desired specifications, allowing for small, unavoidable errors.

Error Propagation

Error propagation refers to how uncertainties in measurements or initial values affect the outcome of a calculation or the final result of an experiment. It's a critical consideration in scientific and engineering measurements. Let's see how this works in practice:

• Whenever quantities are added, subtracted, multiplied, or divided, the errors associated with them also combine according to specific rules.
• In multiplication, as seen in our exercise, the relative tolerances or percentages add up. This is because we expand the multiplication to first order, ignoring higher order terms, since they have a negligible effect at small tolerances.
• To quantify this propagation, consider each measurement's potential error, and understand its impact on the final result. In critical applications, this is important to ascertain the reliability and accuracy of the output.

Learning about error propagation equips students and professionals convincingly measure and report the reliability of their results and decisions.

Interval Arithmetic

Interval arithmetic is a technique used to keep track of errors in numerical computations and ensure reliable results. It involves working with intervals instead of precise numbers, which encompasses all possible values due to rounding or measurement errors within a range.Here's how it can apply:

• By representing a number as a range \[a, b\], where a and b are the lower and upper bounds, it naturally includes the potential error.
• In contrast to traditional arithmetic, where exact numbers are used, interval arithmetic uses intervals for each operation, ensuring that the end result covers all possible outcomes.
• This is particularly useful in scenarios where measurements are subject to uncertainty, or where errors might accumulate through iterative calculations.

In our exercise, using interval arithmetic helps manage the variability introduced by the percentages of the tolerances, effectively capturing the full picture of potential outcomes. This helps ensure our results are both accurate and meaningful, given the inherent uncertainties of the measurements we're dealing with.