Question

The circumference of a circle is measured as 28cm with an error of 0.01cm. The percentage error in the area is, choose the correct option:
A. \[\dfrac{1}{14}\]
B. 0.01
C. \[\dfrac{1}{7}\]
D. none of these

Answer 1

Hint: Use the formula of circumference \[C=2\pi r\] to find the radius (r). Then to find the error, we have to find the differentials of the partial derivative, so here error \[\delta C=0.01\]. Also, \[\delta C=2\pi \,\,\delta r\]. Next find the value of \[\delta r\]and then use the area formula \[A=\pi \,{{r}^{2}}\], Then find the value of \[\delta A\]. Next, the percentage error is area is given as: \[\dfrac{\delta A}{A}\times 100\%\]

Complete step by step answer:
In the question, it is given that the circumference of a circle is measured as 28cm with an error of 0.01cm. Then the percentage error in the area is to be found.

So here we know that the circumference (C) of a circle is given as: \[C=2\pi r\]. Next, we are also given that the circumference is 28 cm, so we have:
\[
  \Rightarrow C=2\pi r \\
  \Rightarrow 28=2\pi r \\
  \Rightarrow r=\dfrac{14}{\pi } \\
\]
Next, when we find the partial derivative of \[C=2\pi r\], we get:
\[\Rightarrow \delta C=2\pi \,\,\delta r\]
So here \[\delta C\]is the error in circumference measurement. Also, we are given that the error circumference is 0.01cm. So, now we have:
\[
 \Rightarrow \delta C=2\pi \,\,\delta r \\
 \Rightarrow 0.01=2\pi \,\,\delta r \\
 \Rightarrow \delta r=\dfrac{0.01}{2\pi \,}\, \\
\]
Now, we will use the area of the circle formula that is given as:
\[\Rightarrow A=\pi {{r}^{2}}\]
Next, we will find the partial derivative as follows:
\[\Rightarrow \delta A=2\pi r\,\delta r\]
Now, since we have to find the percentage error in area so we the percentage error in area given as:
\[
 \Rightarrow \dfrac{\delta A}{A}\times 100 \\
 \Rightarrow \dfrac{2\pi r\delta r}{\pi {{r}^{2}}}\times 100\,\,\,\,\,\,\,\,\because A=\pi {{r}^{2}} \\
 \Rightarrow \dfrac{2\delta r}{r}\times 100 \\
\Rightarrow \dfrac{2\times \dfrac{0.01}{2\pi }}{\dfrac{14}{\pi }}\times 100\,\,\,\,\,\,\,\because r=\dfrac{14}{\pi },\delta r=\dfrac{0.01}{2\pi } \\
  \Rightarrow \dfrac{1}{14} \\
\]

So, the correct answer is “Option A”.

Note: When we are finding the percentage error then make sure you multiply the ratio \[\dfrac{\delta A}{A}\]with 100, so as to get the percentage error. Also, the percentage error of the area is given as \[\dfrac{\delta A}{A}\times 100%\]. The partial derivative is used to find the error and not the percentage error. Also, \[\delta A\] is the error in area measurement.