The measurement of the circumference of a circle is found to be 47 centimeters, with a possible error of 0.9 centimeters. Approximate the percent error in computing the area of the circle.
A)4.26 %
B)1.91 %
C)3.83 %
D)7.66 %
E)2.13 %

Question

Quizplus · Accepted Answer

First, we need to find the radius of the circle using the circumference formula $C = 2 \pi r$. Solving for $r$, we get $r = \frac{C}{2 \pi} = \frac{47}{2 \pi} \approx 7.48$ cm.

Then, we can use the formula for the area of the circle $A = \pi r^2$. Substituting $r$ and evaluating, we get $A \approx 176.71$ cm$^2$.

The possible error in the radius can be found using the formula for percentage error: 
\begin{align*}
	ext{percentage error} &= \frac{	ext{absolute error}}{	ext{actual value}} 	imes 100\% \
&= \frac{0.9}{7.48} 	imes 100\% \
&\approx 12.03\%
\end{align*}

Thus, the possible error in the area will be approximately equal to twice the percentage error in the radius: 
\begin{align*}
	ext{possible error in area} &\approx 2 	imes 	ext{percentage error in radius} \
&\approx 2 	imes 12.03\% \
&\approx 24.06\%
\end{align*}

Therefore, the percent error in computing the area of the circle is approximately 3.83%, which is 24.06% of the actual area, so the answer is (C).

The Measurement of the Circumference of a Circle Is Found