Question

The measurement of the circumference of a circle is found to be 68 centimeters, with a possible error of 0.9 centimeter. (a) Approximate the percent error in computing the area of the circle. (Round your answer to two decimal places

Answer 1

Answer: 2.65%

Step-by-step explanation:

Given : The  measurement of the circumference of a circle =  68 centimeters

Possible error : $dC=0.9$ centimeter.

The formula to find the circumference :-

$C=2\pi r\\\\\Rightarrow\ r=\dfrac{C}{2\pi}\\\\\Rightarrow\ r=\dfrac{68}{2\pi}=\dfrac{34}{\pi}$

Differentiate the formula of circumference w.r.t. r , we get

$dC=2\pi dr\\\\\Rightarrow\ dr=\dfrac{dC}{2\pi}=\dfrac{0.9}{2\pi}=\dfrac{0.45}{\pi}$

The area of a circle  :-

$A=\pi r^2=\pi(\frac{34}{\pi})^2=\dfrac{1156}{\pi}$

Differentiate both sides w.r.t r, we get

$dA=\pi(2r)dr\\\\=\pi(2\times\frac{34}{\pi})(\frac{0.45}{\pi})\\\\=\dfrac{30.6}{\pi}$

The percent error in computing the area of the circle is given by :-

$\dfrac{dA}{A}\times100\\\\\dfrac{\dfrac{30.6}{\pi}}{\dfrac{1156}{\pi}}\times100\\\\=2.64705882353\%\approx 2.65\%$