Answer:
The maximum error in the calculated in the area is about ![27.65 \:cm^2](https://tex.z-dn.net/?f=27.65%20%5C%3Acm%5E2)
The relative error is 0.02
The percentage error is 1.82%
Step-by-step explanation:
(a) Differentials are infinitely small quantities. Given a function
we call
and
differentials and the relationship between them is given by,
![dy=f'(x)dx](https://tex.z-dn.net/?f=dy%3Df%27%28x%29dx)
Let
be the radius of the disk and its area
. If the error in the measured valued of
is denoted by
, then the corresponding error in the calculated value of A is
, which can be approximated by the differential
![dA=(2\pi r)dr](https://tex.z-dn.net/?f=dA%3D%282%5Cpi%20r%29dr)
We know that
and
, substituting into the above differential we get
![dA=(2\pi r)dr\\\\dA=2\pi \cdot 22\cdot 0.2\approx 27.65](https://tex.z-dn.net/?f=dA%3D%282%5Cpi%20r%29dr%5C%5C%5C%5CdA%3D2%5Cpi%20%5Ccdot%2022%5Ccdot%200.2%5Capprox%2027.65)
The maximum error in the calculated in the area is about ![27.65 \:cm^2](https://tex.z-dn.net/?f=27.65%20%5C%3Acm%5E2)
(b) The definition of relative error is
![relative \:error=\frac{absolute \:error}{value \:of \:thing \:measured}](https://tex.z-dn.net/?f=relative%20%5C%3Aerror%3D%5Cfrac%7Babsolute%20%5C%3Aerror%7D%7Bvalue%20%5C%3Aof%20%5C%3Athing%20%5C%3Ameasured%7D)
To find the relative error you need to divide the error by the total area
![\frac{dA}{A}=\frac{(2\pi r)dr}{\pi r^2}=\frac{2dr}{r} =\frac{2\cdot 0.2}{22} \approx 0.02](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7BA%7D%3D%5Cfrac%7B%282%5Cpi%20r%29dr%7D%7B%5Cpi%20r%5E2%7D%3D%5Cfrac%7B2dr%7D%7Br%7D%20%3D%5Cfrac%7B2%5Ccdot%200.2%7D%7B22%7D%20%5Capprox%200.02)
(c) To find the percentage error you need to apply this formula
![percentage\:error=\frac{absolute \:error}{value \:of \:thing \:measured}\times 100\%](https://tex.z-dn.net/?f=percentage%5C%3Aerror%3D%5Cfrac%7Babsolute%20%5C%3Aerror%7D%7Bvalue%20%5C%3Aof%20%5C%3Athing%20%5C%3Ameasured%7D%5Ctimes%20100%5C%25)
![\frac{dA}{A}\times100\%=\frac{(2\pi r)dr}{\pi r^2}\times100\%=\frac{2dr}{r} \times100\%=\frac{2\cdot 0.2}{22} \times100\%\approx1.82\%](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7BA%7D%5Ctimes100%5C%25%3D%5Cfrac%7B%282%5Cpi%20r%29dr%7D%7B%5Cpi%20r%5E2%7D%5Ctimes100%5C%25%3D%5Cfrac%7B2dr%7D%7Br%7D%20%5Ctimes100%5C%25%3D%5Cfrac%7B2%5Ccdot%200.2%7D%7B22%7D%20%5Ctimes100%5C%25%5Capprox1.82%5C%25)