Circumference is

\(\displaystyle{C}={2}\pi{r}\)

Differentiate both sides with respect to r

\(\displaystyle{\frac{{{d}{C}}}{{{d}{r}}}}={2}\pi\)

When \(\displaystyle\triangle{r}\) is small, we can write

\(\displaystyle{\frac{{\triangle{C}}}{{\triangle{r}}}}={2}\pi\)

\(\displaystyle{\frac{{\triangle{C}}}{{{2}\pi}}}=\triangle{r}\)

\(\displaystyle\triangle{r}={\frac{{\triangle{C}}}{{{2}\pi}}}\)

It is given that \(\displaystyle\triangle{C}={0.5}\)

\(\displaystyle\triangle{r}={\frac{{{0.5}}}{{{2}\pi}}}\)

\(\displaystyle\triangle{r}={\frac{{{1}}}{{{4}\pi}}}\)

\(\displaystyle{S}{\left({r}\right)}-{4}\pi{r}^{{{2}}}\)

Differentiate both sides with respect to r

\(\displaystyle{\frac{{{d}{S}}}{{{d}{r}}}}={8}\pi{r}\)

if \(\displaystyle\triangle{r}\) is small, we can say

\(\displaystyle{\frac{{\triangle{S}}}{{\triangle{r}}}}={8}\pi{r}\)

\(\displaystyle\triangle{S}={8}\pi{r}\triangle{r}\)

\(\displaystyle\triangle{S}={4}\cdot{2}\pi{r}\triangle{r}\)

\(\displaystyle\triangle{S}={4}{C}\triangle{r}\)

Where Cis the circumference.

Note that \(\displaystyle\triangle{S}\) will be maximum when \(\displaystyle\triangle{r}\) is maximum. That means, the maximum error in S is:

\(\displaystyle\triangle{S}={4}\cdot{84}\cdot{\frac{{{1}}}{{{4}\pi}}}={\frac{{{84}}}{{\pi}}}\approx{27}\ \text{cm}^{{{2}}}\)

Relative Error: \(\displaystyle={\frac{{\triangle{S}}}{{{S}}}}={\frac{{{\frac{{{84}}}{{\pi}}}}}{{{4}\pi{r}^{{{2}}}}}}={\frac{{{\frac{{{84}}}{{\pi}}}}}{{{4}\pi{\left({\frac{{{C}}}{{{2}\pi}}}\right)}^{{{2}}}}}}={\frac{{{\frac{{{84}}}{{\pi}}}}}{{{4}\pi{\left({\frac{{{84}}}{{{2}\pi}}}\right)}^{{{2}}}}}}={\frac{{{1}}}{{{84}}}}\approx{0.012}\)

\(\displaystyle{C}={2}\pi{r}\)

Differentiate both sides with respect to r

\(\displaystyle{\frac{{{d}{C}}}{{{d}{r}}}}={2}\pi\)

When \(\displaystyle\triangle{r}\) is small, we can write

\(\displaystyle{\frac{{\triangle{C}}}{{\triangle{r}}}}={2}\pi\)

\(\displaystyle{\frac{{\triangle{C}}}{{{2}\pi}}}=\triangle{r}\)

\(\displaystyle\triangle{r}={\frac{{\triangle{C}}}{{{2}\pi}}}\)

It is given that \(\displaystyle\triangle{C}={0.5}\)

\(\displaystyle\triangle{r}={\frac{{{0.5}}}{{{2}\pi}}}\)

\(\displaystyle\triangle{r}={\frac{{{1}}}{{{4}\pi}}}\)

\(\displaystyle{S}{\left({r}\right)}-{4}\pi{r}^{{{2}}}\)

Differentiate both sides with respect to r

\(\displaystyle{\frac{{{d}{S}}}{{{d}{r}}}}={8}\pi{r}\)

if \(\displaystyle\triangle{r}\) is small, we can say

\(\displaystyle{\frac{{\triangle{S}}}{{\triangle{r}}}}={8}\pi{r}\)

\(\displaystyle\triangle{S}={8}\pi{r}\triangle{r}\)

\(\displaystyle\triangle{S}={4}\cdot{2}\pi{r}\triangle{r}\)

\(\displaystyle\triangle{S}={4}{C}\triangle{r}\)

Where Cis the circumference.

Note that \(\displaystyle\triangle{S}\) will be maximum when \(\displaystyle\triangle{r}\) is maximum. That means, the maximum error in S is:

\(\displaystyle\triangle{S}={4}\cdot{84}\cdot{\frac{{{1}}}{{{4}\pi}}}={\frac{{{84}}}{{\pi}}}\approx{27}\ \text{cm}^{{{2}}}\)

Relative Error: \(\displaystyle={\frac{{\triangle{S}}}{{{S}}}}={\frac{{{\frac{{{84}}}{{\pi}}}}}{{{4}\pi{r}^{{{2}}}}}}={\frac{{{\frac{{{84}}}{{\pi}}}}}{{{4}\pi{\left({\frac{{{C}}}{{{2}\pi}}}\right)}^{{{2}}}}}}={\frac{{{\frac{{{84}}}{{\pi}}}}}{{{4}\pi{\left({\frac{{{84}}}{{{2}\pi}}}\right)}^{{{2}}}}}}={\frac{{{1}}}{{{84}}}}\approx{0.012}\)