Question

Find the number of seconds in a year, and the fractional error in assuming that 1yr=π×107 sec

Answer 1

Answer:

There are $3,1536*10^{7}$ seconds in a year.

The fractional error in assuming  $1yr=\pi * 10^{7} sec$  is $3,81*10^{-3}$.

Step-by-step explanation:

Assuming a year of 365 days (non leap-years), we can found how many seconds are in a year by multiplying seconds in a minute, minutes in an hour, hours in a day and finally days in a year.

$Seconds_{year} = 60\frac{seconds}{minute} *60\frac{minutes}{hour}*24\frac{hours}{day}  *365\frac{days}{year}=3,1536*10^{7} \frac{seconds}{year}$

Knowing how many seconds are in a year, we can find the absolute error of the estimation by taking the module of the difference between real value and estimation:

$\varepsilon _{abs} = |real value-estimation|=|3,1536*10^{7} \frac{seconds}{year}-\pi *10^{7} \frac{seconds}{year}|=1,2000734641 *10^{5}\frac{seconds}{year}$

Now that we know the absolute error, we calculate the fractional error by dividing it by the real value:

$\varepsilon _{rel}=\frac{\varepsilon _{abs}}{real value}= \frac{1,2000734641 *10^{5} \frac{seconds}{year}}{3,1536*10^{7} \frac{seconds}{year}}=3,81 *10^{-3}$

That is a minimal error, close to 0,4% and enough for every day estimations.