Question

The radius of a circular disk is given as 22 cm with a maximum error in measurement of 0.2cm.

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

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

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