Answer: 2.65%
Step-by-step explanation:
Given : The measurement of the circumference of a circle = 68 centimeters
Possible error : [tex]dC=0.9[/tex] centimeter.
The formula to find the circumference :-
[tex]C=2\pi r\\\\\Rightarrow\ r=\dfrac{C}{2\pi}\\\\\Rightarrow\ r=\dfrac{68}{2\pi}=\dfrac{34}{\pi}[/tex]
Differentiate the formula of circumference w.r.t. r , we get
[tex]dC=2\pi dr\\\\\Rightarrow\ dr=\dfrac{dC}{2\pi}=\dfrac{0.9}{2\pi}=\dfrac{0.45}{\pi}[/tex]
The area of a circle :-
[tex]A=\pi r^2=\pi(\frac{34}{\pi})^2=\dfrac{1156}{\pi}[/tex]
Differentiate both sides w.r.t r, we get
[tex]dA=\pi(2r)dr\\\\=\pi(2\times\frac{34}{\pi})(\frac{0.45}{\pi})\\\\=\dfrac{30.6}{\pi}[/tex]
The percent error in computing the area of the circle is given by :-
[tex]\dfrac{dA}{A}\times100\\\\\dfrac{\dfrac{30.6}{\pi}}{\dfrac{1156}{\pi}}\times100\\\\=2.64705882353\%\approx 2.65\%[/tex]
