Chapter 1, Problem 34GP

To determine

The percent uncertainty and required number of significant digits to express distance from the satellite.

Answer to Problem 34GP

Solution :

The required uncertainty in the distance is ${10}^{-5}\%$ . The required number of significant digits is 8.

Explanation of Solution

Given :

The satellite is 20,000 km away.

The desired accuracy is $\pm 2\text{ m}$ .

Formula Used:

$\text{Percentage uncertainty = }\frac{\text{|Required accuracy|}}{\text{Distance}}\times 100\text{\%}$

Calculation:

Convert the unit of distance from kilometer to meter.

The distance from the satellite is

  $\begin{array}{l}d=20,000\text{ km}\end{array}$

  $\begin{array}{c}d=20,000\times {10}^3\text{ m}\end{array}$

  $\begin{array}{c}d=2\times {\text{10}}^7\text{m}\text{.}\end{array}$

Percentage uncertainty $=\frac{2}{2\times {10}^7}\times 100\text{ \%}$

         $={10}^{-5}\%$

The required uncertainty in the distance is ${10}^{-5}\%$ .

The satellite is $\text{20},\text{000,000 m}$ away and the desired accuracy is $\pm 2\text{ m}$ . It means the value can vary from 19,999,998 m to 20,000,002 m. Significant digits are the number of digits which represent the required accurate number. Thus, the required number of significant digits is 8.

Conclusion:

The required uncertainty in the distance is ${10}^{-5}\%$ . The number of significant digits required is 8.