Question

The radius of a circular disk is given as 22 cm with a maximum error in measurement of 0.2cm.
(a) Use differentials to estimate the maximumerror in the calculated area of the disk. (Round the answer to twodecimal places.)
cm2

(b) What is the relative error? (Round the answer to four decimalplaces.)

What is the percentage error? (Round the answer to two decimalplaces.)
%

Answer 1

Answer:

The maximum error in the calculated in the area is about $27.65 \:cm^2$

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 $y=f(x)$ we call $dy$ and $dx$ differentials and the relationship between them is given by,

$dy=f'(x)dx$

Let $r$ be the radius of the disk and its area $A=\pi r^2$. If the error in the measured valued of $r$ is denoted by $dr=\Delta r$, then the corresponding error in the calculated value of A is $\Delta A$, which can be approximated by the differential

$dA=(2\pi r)dr$

We know that $r = 22$ and $dr=0.2$, substituting into the above differential we get

$dA=(2\pi r)dr\\\\dA=2\pi \cdot 22\cdot 0.2\approx 27.65$

The maximum error in the calculated in the area is about $27.65 \:cm^2$

(b) The definition of relative error is

$relative \:error=\frac{absolute \:error}{value \:of \:thing \:measured}$

To find the relative error you need to divide the error by the total area

$\frac{dA}{A}=\frac{(2\pi r)dr}{\pi r^2}=\frac{2dr}{r} =\frac{2\cdot 0.2}{22} \approx 0.02$

(c) To find the percentage error you need to apply this formula

$percentage\:error=\frac{absolute \:error}{value \:of \:thing \:measured}\times 100\%$

$\frac{dA}{A}\times100\%=\frac{(2\pi r)dr}{\pi r^2}\times100\%=\frac{2dr}{r} \times100\%=\frac{2\cdot 0.2}{22} \times100\%\approx1.82\%$