Answer
$\sqrt 8.05^{2}+5.97^{2} \approx 10 + 0.022= 10.022$
Calculator gives $\sqrt 8.05^{2}+5.97^{2}\approx10.0221$
The error is approximately 0.0001
Work Step by Step
We are given: $\sqrt 8.05^{2}+5.97^{2}$
Let $f(x,y)=\sqrt x^{2}+y^{2}$ with $x=8, dx=0.05, y=6, dy=-0.03$
then use dz to approximate $\Delta z$
$dz=f_{x}(x,y).dx+d_{y}(x,y).dy$
$dz=(\frac{1}{2\sqrt x^{2}+y^{2}}.2x)dx+(\frac{1}{2\sqrt x^{2}+y^{2}}.2y)dy$
$dz=(\frac{x}{\sqrt x^{2}+y^{2}})dx+(\frac{y}{\sqrt x^{2}+y^{2}})dy$
$dz=\frac{4}{5}(0.05)+\frac{3}{5}(-0.03)$
$dz=0.022$
Thus, $\sqrt 8.05^{2}+5.97^{2} \approx 10 + 0.022= 10.022$
Calculator gives $\sqrt 8.05^{2}+5.97^{2}\approx10.0221$
The error is approximately 0.0001