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Hint: Here we have to apply the formulas:

Area of the rectangle = length $\times $ breadth

Perimeter of the rectangle = 2(length + breadth)

Time = $\dfrac{\text{Distance}}{\text{speed}}$

Complete step-by-step answer:

Here, the area of the rectangular field is given as, $A=3584\text{ }{{m}^{2}}$

Length of the rectangular field is, $l=64\text{ }m$.

First, we have to find the breadth of the rectangular field, $b$.

We know the formula that the area of the rectangle,

$A=l\times b$

$3584=64\times b$

Therefore, by cross multiplication we get:

$\dfrac{3584}{64}=b$

By cancellation we get:

$b=56$

Hence, the breadth of the rectangle, $b=56$.

Next, we have to find the perimeter of the rectangular field, $P$. i.e.

$P=2(l+b)$

We have, $l=64$ and $b=56$. Therefore we get:

$\begin{align}

& P=2(64+56) \\

& P=2\times 120 \\

& P=240 \\

\end{align}$

Hence, we got the perimeter of the rectangular field as, $P=240\text{ }m$.

We are given that the boy runs 5 rounds of field.

Therefore, the perimeter for 5 rounds = $240\times 5=1200\text{ }m$

It is also given that the speed of the boy running around the field, $S=6\text{ }km/hr$

The speed is given in $km/hr$, now we have to change it into $m/s$ by multiplying it with $\dfrac{5}{18}$.

Therefore, the speed,

$S=6\times \dfrac{5}{18}$

By cancellation we get:

$S=\dfrac{5}{3}m/s$

Next, we have to find the time required to cover 5 rounds of field, i.e. the time required to cover the distance of $1200\text{ }m$. Hence we get:

Time, $T=\dfrac{D}{S}$ , where $D$ is the distance and $S$ is the speed.

$T=\dfrac{1200}{\dfrac{5}{3}}$

We know that $\dfrac{a}{\dfrac{b}{c}}=\dfrac{ac}{b}$. Therefore, we get,

$T=\dfrac{1200\times 3}{5}$

By cancellation we get:

$\begin{align}

& T=240\times 3 \\

& T=720 \\

\end{align}$

We got time, $T=720$ seconds. Now we have to convert it into minutes by dividing it with 60.

$T=\dfrac{720}{60}$

By cancellation we get:

$T=12$ min

Hence, the time required to cover 5 rounds of field is 12 min.

Note: Here we have to calculate the perimeter of 5 rounds of field. So we have to multiply the perimeter by 5. In the question the unit is given in metre, so donâ€™t forget to change the unit from $km/hr$ to $m/s$.

Area of the rectangle = length $\times $ breadth

Perimeter of the rectangle = 2(length + breadth)

Time = $\dfrac{\text{Distance}}{\text{speed}}$

Complete step-by-step answer:

Here, the area of the rectangular field is given as, $A=3584\text{ }{{m}^{2}}$

Length of the rectangular field is, $l=64\text{ }m$.

First, we have to find the breadth of the rectangular field, $b$.

We know the formula that the area of the rectangle,

$A=l\times b$

$3584=64\times b$

Therefore, by cross multiplication we get:

$\dfrac{3584}{64}=b$

By cancellation we get:

$b=56$

Hence, the breadth of the rectangle, $b=56$.

Next, we have to find the perimeter of the rectangular field, $P$. i.e.

$P=2(l+b)$

We have, $l=64$ and $b=56$. Therefore we get:

$\begin{align}

& P=2(64+56) \\

& P=2\times 120 \\

& P=240 \\

\end{align}$

Hence, we got the perimeter of the rectangular field as, $P=240\text{ }m$.

We are given that the boy runs 5 rounds of field.

Therefore, the perimeter for 5 rounds = $240\times 5=1200\text{ }m$

It is also given that the speed of the boy running around the field, $S=6\text{ }km/hr$

The speed is given in $km/hr$, now we have to change it into $m/s$ by multiplying it with $\dfrac{5}{18}$.

Therefore, the speed,

$S=6\times \dfrac{5}{18}$

By cancellation we get:

$S=\dfrac{5}{3}m/s$

Next, we have to find the time required to cover 5 rounds of field, i.e. the time required to cover the distance of $1200\text{ }m$. Hence we get:

Time, $T=\dfrac{D}{S}$ , where $D$ is the distance and $S$ is the speed.

$T=\dfrac{1200}{\dfrac{5}{3}}$

We know that $\dfrac{a}{\dfrac{b}{c}}=\dfrac{ac}{b}$. Therefore, we get,

$T=\dfrac{1200\times 3}{5}$

By cancellation we get:

$\begin{align}

& T=240\times 3 \\

& T=720 \\

\end{align}$

We got time, $T=720$ seconds. Now we have to convert it into minutes by dividing it with 60.

$T=\dfrac{720}{60}$

By cancellation we get:

$T=12$ min

Hence, the time required to cover 5 rounds of field is 12 min.

Note: Here we have to calculate the perimeter of 5 rounds of field. So we have to multiply the perimeter by 5. In the question the unit is given in metre, so donâ€™t forget to change the unit from $km/hr$ to $m/s$.