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# The radius of a circular disc is given as $24cm$ with a maximum error in mesurement of $0.02cm$ Use differentials to estimate the maximum error in the calculated area of the dise.

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Toolbox:
• Let $y=f(x)$ be a differentiable function then the quantities $dx$ and $dy$ are called differentials.The differential $dx$ is an independent variable.
• The differential $dy$ is then defined by $dy=f'(x)dx(dx \approx \Delta x)$
• Also $f(x+\Delta x)-f(x)=\Delta y =dy$ from which $f(x+\Delta x)$can be evaluated
• $\large\frac{\Delta y}{y}=\frac{Actaual\;change\;in \;y}{Actual\;value\;of \;y}$$=relative \;error • \large\frac{\Delta y}{y}$$ \times 100 =percentage\; error$
The area of the disc is $A=\pi r^2$ where
x is the radius
$\therefore dA=2 \pi x dx$
When $x=24cm, dx \approx \Delta x =0.02$
$\therefore dA =2 \pi \times 24 \times 0.02$
$\qquad=0.96 \pi cm^2$
answered Aug 12, 2013 by

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