\magnification=\magstep1

$$
C \quad = \quad \sqrt{A^2 + B^2} 
$$

$$
\eqalign{
A &= \phantom{1}5.3 \pm 0.2 \ {\rm cm}  \cr
B &= 15.4 \pm 0.3 \ {\rm cm} \cr
}
$$


$$
{\rm if \qquad } C \ = \ \sqrt{A^2 + B^2} 
       {\rm \quad , \quad what\ is\ \quad } \Delta C?
$$

$$
\eqalign{
{{ \Delta A^2 }\over{A^2} } \quad &= \quad 
          {{\Delta A}\over{A}} + {{\Delta A}\over{A}} \cr
   &= \quad {{0.2}\over{5.3}} + {{0.2}\over{5.3}}  \cr
   &= \quad 0.075 \cr
}
$$

$$
{{\Delta A^2}} \quad = \quad 0.075 A^2  \ = \ 2.1 {\rm \ cm^2}
$$

$$
A^2 \quad = \quad 28.1 \pm 2.1 {\rm \ cm^2 }
$$

$$
B^2 \quad = \quad 237.2 \pm 9.2 {\rm \ cm^2 }
$$


$$
C^2 \quad = \quad {A^2 + B^2} \quad = \quad 265.3 \pm 11.3 {\rm \ cm^2 }
$$


$$
{{\Delta C^2}\over{C^2}} \quad = \quad {{11.3}\over{265.3}} \ = \ 0.043 
$$

$$
\eqalign{
{{\Delta C}\over{C}} \quad &= \quad {{1}\over{2}} {{\Delta C^2}\over{C^2}} \cr
                           &= \quad 0.021 \cr
}
$$

$$
{\Delta C} \quad = \quad 0.02 \thinspace C 
                        \ = \ 0.021 \thinspace (16.3 {\rm \ cm})  
                        \ = \ 0.3 {\rm \ cm}
$$

$$
{C} \quad = \quad 16.3 \pm 0.3 {\rm \ cm}
$$

\bye
