Answer

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Hint: We will use the derivative method to find the error percentage in the area. Also, we will use the formula of circumference of circle and area of the area which is given by.

$\begin{align}

& Circumference=2\pi r \\

& Area=\pi {{r}^{2}} \\

\end{align}$

Where ‘r’ is the radius of the circle.

We have been given the circumference of a circle as 56cm with an error of 0.02cm.

We know that circumference of a circle is equal to $2\pi r$, where ‘r’ is the radius of the circle.

Let us suppose the radius of the given circle is ‘r’ cm.

$\begin{align}

& \Rightarrow 2\pi r=56 \\

& \Rightarrow \pi r=28 \\

& \Rightarrow r=\dfrac{28}{\pi } \\

\end{align}$

Circumference (C) $=2\pi r$

On differentiating both sides, we get

$\dfrac{dc}{dr}=2\pi .........\left( 1 \right)$

Now, we know that area of a circle is equal to $\pi {{r}^{2}}$, where ‘r’ is the radius of the circle.

$\Rightarrow A=\pi {{r}^{2}}$

On differentiating with respect of ‘r’ we get,

$\dfrac{dA}{dr}=\dfrac{d\left( \pi {{r}^{2}} \right)}{dr}$

Since, $'\pi '$ is a constant.

$=\pi \dfrac{d{{r}^{2}}}{dr}$

As, we know that $\dfrac{d{{x}^{n}}}{dx}=n\times {{x}^{n-1}}$

$\Rightarrow \dfrac{dA}{dr}=2\pi r.........\left( 2 \right)$

Now, dividing equation (2) by (1), we get,

$\begin{align}

& \dfrac{dA}{dc}=r \\

& \Rightarrow dA=rdc \\

\end{align}$

Here, dA is the error in Area and dc is the error in circumference.

$\Rightarrow dA=r\left( 0.02 \right)$

On dividing by $\pi {{r}^{2}}$ we get,

$\begin{align}

& \dfrac{dA}{\pi {{r}^{2}}}=\dfrac{r\left( 0.02 \right)}{\pi {{r}^{2}}} \\

& \dfrac{dA}{A}=\dfrac{0.02}{\pi r} \\

\end{align}$

We have $\pi r=28$.

$\begin{align}

& \Rightarrow \dfrac{dA}{A}=\dfrac{0.02}{28} \\

& \Rightarrow \dfrac{dA}{A}=\dfrac{1}{1400} \\

\end{align}$

So, percentage error $=\dfrac{dA}{A}\times 100=\dfrac{1}{1400}\times 100$

$=\dfrac{1}{14}%$

Therefore, the correct option of the given question is option (C).

Note: Remember that dA means the error in the function ‘A’ and on dividing it by A we get the relative error. Also, be careful while doing calculation and differentiation while finding the error and substitute the value carefully. Sometimes the student might miss the ‘2’ in the formula for circumference and this might lead to the wrong answer.

$\begin{align}

& Circumference=2\pi r \\

& Area=\pi {{r}^{2}} \\

\end{align}$

Where ‘r’ is the radius of the circle.

__Complete step-by-step answer:__We have been given the circumference of a circle as 56cm with an error of 0.02cm.

We know that circumference of a circle is equal to $2\pi r$, where ‘r’ is the radius of the circle.

Let us suppose the radius of the given circle is ‘r’ cm.

$\begin{align}

& \Rightarrow 2\pi r=56 \\

& \Rightarrow \pi r=28 \\

& \Rightarrow r=\dfrac{28}{\pi } \\

\end{align}$

Circumference (C) $=2\pi r$

On differentiating both sides, we get

$\dfrac{dc}{dr}=2\pi .........\left( 1 \right)$

Now, we know that area of a circle is equal to $\pi {{r}^{2}}$, where ‘r’ is the radius of the circle.

$\Rightarrow A=\pi {{r}^{2}}$

On differentiating with respect of ‘r’ we get,

$\dfrac{dA}{dr}=\dfrac{d\left( \pi {{r}^{2}} \right)}{dr}$

Since, $'\pi '$ is a constant.

$=\pi \dfrac{d{{r}^{2}}}{dr}$

As, we know that $\dfrac{d{{x}^{n}}}{dx}=n\times {{x}^{n-1}}$

$\Rightarrow \dfrac{dA}{dr}=2\pi r.........\left( 2 \right)$

Now, dividing equation (2) by (1), we get,

$\begin{align}

& \dfrac{dA}{dc}=r \\

& \Rightarrow dA=rdc \\

\end{align}$

Here, dA is the error in Area and dc is the error in circumference.

$\Rightarrow dA=r\left( 0.02 \right)$

On dividing by $\pi {{r}^{2}}$ we get,

$\begin{align}

& \dfrac{dA}{\pi {{r}^{2}}}=\dfrac{r\left( 0.02 \right)}{\pi {{r}^{2}}} \\

& \dfrac{dA}{A}=\dfrac{0.02}{\pi r} \\

\end{align}$

We have $\pi r=28$.

$\begin{align}

& \Rightarrow \dfrac{dA}{A}=\dfrac{0.02}{28} \\

& \Rightarrow \dfrac{dA}{A}=\dfrac{1}{1400} \\

\end{align}$

So, percentage error $=\dfrac{dA}{A}\times 100=\dfrac{1}{1400}\times 100$

$=\dfrac{1}{14}%$

Therefore, the correct option of the given question is option (C).

Note: Remember that dA means the error in the function ‘A’ and on dividing it by A we get the relative error. Also, be careful while doing calculation and differentiation while finding the error and substitute the value carefully. Sometimes the student might miss the ‘2’ in the formula for circumference and this might lead to the wrong answer.

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