Question

How do you use dimensional analysis to figure out how many seconds are in 4 years?

Answer 1

Hint: We are given 4 years, we are asked to find numbers of seconds are there in it and we have to do it by dimensional analysis, to do so we will learn what does a dimensional analysis means that we will learn how are 4 year and the seconds can be related, we will use that 1 year is same as 365 days then we use 1 day is same as 24 hours then we use that 1 hour is same as 60 minutes and lastly we use that 1 minute is same as 60 seconds so we will combine all these dimension to find the number of seconds in 4 years.

Complete step by step answer:
We are given 4 years; we have to change it into the number of seconds.
Before this we will learn what dimensional analysis meant,
Now dimensional analysis also known as unit factor method is a problem solving method that uses the fact that any number or expression can be multiplied by one without changing its value. It is a great technique to change the dimension of one quantity, quantity remains the same but just the dimension will get changed.
For example: we know 1 meter has 100 centimeter
So, $1m=1cm$ , the quantity is the same , just the dimension is different.
Now, we are given that we have 4 years, that is we have time in a year, we have to find the equivalent time in the seconds.
To do so we will learn how the different dimensions of the time related things are connected.
1 year = 365 days …………………………………… (1)
In 1 day = 24 hours ………………………………… (2)
1 hour = 60 minutes ………………………………. (3)
1 minute = 60 seconds ………………………….. (4)
We have to start from this year and reach it in seconds.
So, now –
$4\text{years}=4\times 1$
From eq (1) we get –
$1=\dfrac{365\text{days}}{1\text{year}}$
$=4\text{year}\times \dfrac{365\text{days}}{1\text{year}}$
From eq (2), we get –
$1=\dfrac{24\text{hours}}{1\text{day}}$
So, $4\text{year}\times \dfrac{365\text{days}}{1\text{year}}\times \dfrac{24\text{hours}}{1\text{day}}$
Now from eq (3) we get –
$1=\dfrac{60\text{minutes}}{1\text{hour}}$
So,
$=4\text{year}\times \dfrac{365\text{days}}{1\text{year}}\times \dfrac{24\text{hour}}{1\text{day}}\times \dfrac{60\text{minutes}}{1\text{hour}}$
From eq (4), we get –
$1=\dfrac{60\text{seconds}}{1\text{minutes}}$
So,
$=4\text{year}\times \dfrac{365\text{days}}{1\text{year}}\times \dfrac{24\text{hour}}{1\text{day}}\times \dfrac{60\text{minutes}}{1\text{hour}}\times \dfrac{60\text{seconds}}{1\text{minute}}$
Now we cancel like terms, we get –
$=4\times 365\times 24\times 60\times 60\text{seconds}$
By simplifying, we get –
$4\text{year=12614400seconds}$

Note: Remember that we need to be careful while multiplying as when we change a big unit to the small then it will get a little larger to calculate. So error may arise also we cannot skip any in between units, we always need to change the term unit by unit. It is widely used in chemistry to change the number of atoms to mole or change large units into small ones.