Question

The area of a square field is $60025{{{m^2}}}$. A man cycles along its boundary at $18{\text{km/hr}}$. In how much time will he return to the starting point?

Answer 1

Hint:
Here, we have to find the time taken to reach the starting point. First, we will find the length of the square by using the area of the square, and then we will find the perimeter to find the distance of the boundary. Then we will use the speed formula to find the time taken. Speed is defined as the ratio of the distance traveled by the time taken.

Formula Used:
We will use the following formula:
1) Area of a square is given by the formula ${\text{Area}} = {a^2}$ where $a$ is the length of the square.
2) Perimeter of a square is given by the formula ${\text{Perimeter}} = 4a$, where $a$ is the length of the square.
3) Speed is given by the formula ${\text{Speed}} = \dfrac{{{\text{Distance}}}}{{{\text{Time}}}}$

Complete step by step solution:
According to the question,
Area of a square field $ = 60025{{m^2}}$
Substituting ${\text{Area}} = {a^2}$ in the above equation, we get
\[ \Rightarrow {a^2} = 60025\]
Taking square root on both the sides, we get
\[ \Rightarrow a = \sqrt {60025}\]
Now, we will be using the method of prime factorization to find the square root.
$\begin{array}{*{20}{l}}
  5| {60025} \\
\hline
  5| {12005} \\
\hline
  7| {2401} \\
\hline
  7| {343} \\
\hline
  7| {49} \\
\hline
  7| 7 \\
\hline
  {}| 1
\end{array}$
Thus, rewriting 60025 as $5 \times 5 \times 7 \times 7 \times 7 \times 7$ we get
\[ \Rightarrow a = \sqrt {5 \times 5 \times 7 \times 7 \times 7 \times 7} \]
As we are computing square roots so we will pick out only one number from the above pairs. Therefore, we get
\[ \Rightarrow a = 5 \times 7 \times 7\]
Multiplying the terms, we get
\[ \Rightarrow a = 245m\]
Thus, the square is of length 245 m.
Since a man cycles along its boundary, we will find the perimeter of the square.
Perimeter of a square $ = 4a$
Substituting \[a = 245m\] in the above equation, we get
$ \Rightarrow $ Perimeter of a square $ = 4 \times 245$
Multiplying the terms, we get
$ \Rightarrow $ Perimeter of a square$ = 980m$
We are given that the speed of the man is $18{\text{km/hr}}$.
Since the speed is given in ${\text{km/hr}}$, we will convert it into ${\text{m/s}}$.
We know that $1{\text{km}} = 1000{\text{m}}$, $1{\text{hr}} = 3600{\text{s}}$. Therefore,
${\text{Speed}} = \dfrac{{18 \times 1000}}{{3600}}{\text{m/s}}$
By dividing the terms, we get
$ \Rightarrow {\text{Speed}} = \dfrac{{18 \times 5}}{{18}}{\text{m/s}}$
 $ \Rightarrow {\text{Speed}} = 5{\text{m/s}}$
From the speed formula we can get
${\text{Time}} = \dfrac{{{\text{Distance}}}}{{{\text{Speed}}}}$
We know that here the distance is the perimeter of the square field.
By substituting the values of distance and speed, we get
$ \Rightarrow {\text{Time}} = \dfrac{{980}}{5}$
Dividing the terms, we get
$ \Rightarrow {\text{Time}} = {\text{196}}$ seconds
Since $1{\text{hr}} = 60{\text{s}}$, we get
$ \Rightarrow {\text{Time}} = \dfrac{{196}}{{60}}{\text{ hr}}$
Dividing the terms, we get
$ \Rightarrow {\text{Time}} = {\text{3}}\dfrac{{16}}{{60}}{\text{ hr}}$
$ \Rightarrow {\text{Time}} = {\text{3hr16 min}}$
Thus the time taken by the man is 3 hours and 16 minutes.

Therefore, a man takes 3 hours and 16 minutes to reach the starting point again.

Note:
We need to keep in mind that the boundary of the square field is equal to the perimeter of the square. Here, the units of all the different quantities are different and we cannot mathematically operate quantities with two different units. Therefore, we converted the unit of speed from ${\text{km/hr}}$ to ${\text{m/s}}$. We can convert an hour into seconds by multiplying it by 3600. In an hour, there are 60 minutes and 3600 seconds.