Chapter 1, Problem 10P

To determine

The area and its approximate uncertainty of a circle.

Answer to Problem 10P

Solution:

The area of the circle is $A=3.0\times {10}^9{\text{cm}}^{\text{2}}$

Approximate uncertainty is $\text{6}\text{.4\%}$ .

Explanation of Solution

Given:

The radius of the circle is $3.1\times {10}^4\text{cm}$ .

Formula Used:

Area of the circle can be obtained by: $\text{A=}\pi {\text{r}}^2$

Where, r is the radius.

  $\text{Percent uncertainty = }\frac{\text{Absolute uncertainty}}{\text{Absolute value}} \times 100 \%$

Calculation:

Substitute the values:

  $\begin{array}{l}A=\pi r^2\\ A=\left(3.14\right){\left(3.1\times {10}^4\text{cm}\right)}^2\\ A=3.01754\times {10}^9{\text{cm}}^{\text{2}}\end{array}$

Rounding off to least significant digits: $A=3.0\times {10}^9{\text{cm}}^{\text{2}}$

  $\text{Percent uncertainty = }\frac{\text{Absolute uncertainty}}{\text{Absolute value}} \times 100 \%$

From theory of errors:

  $\begin{array}{l}A=\pi r^2\\ \text{ln A= ln}\left({\text{πr}}^{\text{2}}\right)\\ \text{ln A= ln π + 2 ln r}\\ \frac{\Delta A}{A}=2\frac{\Delta r}{r}\\ \frac{\Delta A}{A}=2\times \frac{0.1\times {10}^4}{3.1\times {10}^4}\\ \frac{\Delta A}{A}=0.064\\ \text{Percentage uncertainty =0}\text{.064}\times \text{100\%=6}\text{.4\%}\end{array}$

Conclusion:

The area of the circle is $A=3.0\times {10}^9{\text{cm}}^{\text{2}}$

Approximate uncertainty is $\text{6}\text{.4\%}$ .