Question

The radius of a circular disk is given as 27 cm with a maximum error in measurement of 0.2 cm.
1) Use differentials to estimate the maximum error in the calculated area of the disk. (Round your answer to two decimal places.)
2)What is the relative error? (Round your answer to four decimal places.)
3)What is the percentage error? (Round your answer to two decimal places.)

Answer 1

The maximum error in the calculated area of the disk is 33.93 square cm. The relative error is 0.0148. The percentage error is 1.48%.

a. The area $\mathrm{A}$ of a circle with radius $\mathrm{r}$ is given by the following formula from Geometry:

$A=\pi r^2$

The question provides us the following information:

radius $r=27 \mathrm{~cm}$, maximum radius measurement error $\Delta r=0.2 \mathrm{~cm}$

Using linear differentials we get the error in the calculation of the area of this disk $\Delta A$ from (1) as follows:

$\Delta A=2 \pi r \Delta r \Longrightarrow|\Delta A|=2 \pi r|\Delta r|$

Using the information from (2) in (3) yields

$|\Delta A|=2 \pi(27)(0.2)=33.93 \mathrm{~cm}^2 \text {. }$

The maximum error in the calculated area of the disk is 33.93 square $\mathrm{cm}$.

b. The relative error is calculated using (1) and $r=27 \mathrm{~cm}$ as follows:

Relative Error $=\frac{|\Delta A|}{A}=\frac{33.93}{\pi\left(27^2\right)}=0.0148$.

The relative error is 0.0148.

c. The percentage error is the relative error in part b. above multiplied by $100 \%$ to give us

Percentage Error $=100 \% \times$ Relative Error $=100 \% \times 0.0148=1.48 \%$.

The percentage error is $1.48 \%$.

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