Question

the radius of a circular disk is given as 30 cm with a maximum error in measurement of cm. use differentials to estimate the maximum possible error in the calculated area of the disk.

Answer 1

The calculated area's maximum percentage relative error is approximately 1.33%.

we know that:

Area of the circular disc is given by A = πr²

consider A = πr²

Differentiate both sides with respect to r, we get

dA/dr = 2πr

⇒ dA = 2πr dr

calculate area:

A = πr²

A = π(30)²

A = 900π

A = 2826 cm²

Because of an error in measurement, the radius might actually be as big as 30+0.2 cm

= 30.2 cm

If r is increased from 30 by an amount Δr = dr = 0.2

Then the actual change in the calculated area would be :

ΔA = A(30+Δr)-A(30)

ΔA = A(30.2)-A(30)

ΔA = (30.2)²π -  900π

ΔA = 912.04π - 900π

= 12.04π cm²

So we estimated the maximum error in calculated area as 12.04π cm²

≅ 37.8 cm²

The maximum relative error in calculated area is:

ΔA/A= dA/A = 12.04π/900π

= 0.013

= 1.33 %

The maximum % relative error in calculated area is about 1.33%.

Learn more about Differentials here:

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