Worked exercise — Area of a measured rectangle

Problem

A rectangle has its sides measured as intervals: height $h = [1{,}3\;;\;1{,}4]\;\text{cm}$ and base $b = [6{,}2\;;\;6{,}3]\;\text{cm}$. What is the interval within which its area falls?

Solution

Step 1 — Interval arithmetic. The minimum area is obtained by multiplying the minimum values of the two sides, the maximum area by multiplying the maximum values.

$A_\text{min} = 1{,}3\cdot 6{,}2 = 8{,}06\;\text{cm}^2 \qquad A_\text{max} = 1{,}4\cdot 6{,}3 = 8{,}82\;\text{cm}^2$

Step 2 — Interval of the area. The area therefore lies in the interval:

$\ev{A = [\,8{,}06\;;\;8{,}82\,]\;\text{cm}^2}$

equivalent to the notation $8{,}0(6)\;\div\;8{,}8(2)\;\text{cm}^2$.

Step 3 — Relative uncertainty. The area is known with an uncertainty of about $0{,}76\;\text{cm}^2$ on a mean value of about $8{,}4\;\text{cm}^2$, hence a relative uncertainty of $\sim 9\%$. Note how the relative uncertainty of the area (roughly the sum of the relative uncertainties of the two sides) is larger than that of the individual sides.

Linked atoms

Argomenti: Physical quantities and measurement Concetti: Significant figures and uncertainty Competenze: Use of significant figures