Question

The area of a square field is 8 hectares. How long would a man take to cross it diagonally by walking at the rate of 4 kmph?

Answer 1

Hint: We start solving the problem by drawing the figure representing the given information and assuming the length of the side of the square field. We then recall the conversion that 1 hectare = $0.01k{{m}^{2}}$ to find the area of the field in $k{{m}^{2}}$. We then use the fact that the area of the square with side ‘a’ is ${{a}^{2}}$ to find the length of the side of the square field. We then find the length of the diagonal using the fact that the length of the diagonal of the square with side ‘a’ is $a\sqrt{2}$. We then find the time required to cross the diagonal by using the fact that time = $\dfrac{\text{distance}}{\text{speed}}$.

Complete step by step answer:
According to the problem, we are given that the area of the square field is 8 hectares. We need to find the time taken by the person to cross the field diagonally by walking at a speed of 4 kmph.
Let us draw the figure representing the given information.

We know that 1 hectare = $0.01k{{m}^{2}}$. So, we get 8 hectares = $0.08=\dfrac{2}{25}k{{m}^{2}}$ ---(1).
Let us assume the length of the side of the square field to be ‘a’ km. We know that the area of the square with side ‘a’ is ${{a}^{2}}$.
So, we get ${{a}^{2}}=\dfrac{2}{25}k{{m}^{2}}$.
$\Rightarrow a=\dfrac{\sqrt{2}}{5}km$.
We know that the length of the diagonal of a square with side ‘a’ is $a\sqrt{2}$.
So, the length of diagonal of square field is $a\sqrt{2}=\dfrac{\sqrt{2}}{5}\times \sqrt{2}=\dfrac{2}{5}km$.
We need to find the time to cross the field diagonally at a speed of 4 kmph.
We know that time = $\dfrac{\text{distance}}{\text{speed}}$.
So, we get time = $\dfrac{\dfrac{2}{5}}{4}=\dfrac{1}{10}hours$.
We know that 1 hour = 60 minutes. So, we get time = $\dfrac{1}{10}\times 60=6$ minutes.

∴ The time taken for a man take to cross it diagonally by walking at the rate of 4 kmph is 6 minutes.

Note: We should not make calculation mistakes while solving this problem. We should not stop solving the problem after finding the time in hours as expressing time in a fraction of hours leads to confusion. Whenever we get this type of problem, we should first draw the figure representing the given information. Similarly, we can expect the problems to find the time taken to cross a rectangular field diagonally with ratio of length to breadth as 2:1 having the same area.