Question

1 - 13. a useful and easy - to - remember approximate value for the number of seconds in a year is $pi\times10^{7}$. determine the percent error in this approximate value. (there are 365.24 days in one year.) answer: we all know that in one year = 365.24 days one day = 24 hours one hour = 60 minutes one minute = 60 seconds $x_{exact}=\frac{365.24}{days}\times\frac{24 hours}{day}\times\frac{60 mins}{hour}\times\frac{60 sec}{min}=31,556,936 seconds$ $x_{approx}=pi\times10^{7}=31,415,926.54 seconds$ calculate % error $% error=\frac{|x_{approx}-x_{exact}|}{x_{exact}}\times100%$ $% error=\frac{|31,415,926.54 - 31,556,936|}{31,556,936}\times100%$ $% error = 0.45%$

Explanation:

Step1: Calculate exact number of seconds in a year

We know 1 year = 365.24 days, 1 day = 24 hours, 1 hour = 60 minutes, 1 minute = 60 seconds. So $X_{exact}=365.24\times24\times60\times60 = 31556976$ seconds.

Step2: Calculate approximate number of seconds

Given $X_{approx}=\pi\times 10^{7}\approx3.14159\times 10^{7}=31415926.54$ seconds.

Step3: Calculate percent - error

The formula for percent - error is $\text{Percent Error}=\frac{\vert X_{approx}-X_{exact}\vert}{X_{exact}}\times 100\%$. Substitute the values: $\text{Percent Error}=\frac{\vert31415926.54 - 31556976\vert}{31556976}\times 100\%=\frac{141049.46}{31556976}\times 100\%\approx0.45\%$.

Answer:

$0.45\%$