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Q)

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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A)
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  • 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$
$\therefore dA =2 \pi \times 24 \times 0.02$
$\qquad=0.96 \pi cm^2$
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