A rectangular field has its length and breadth in the ratio of $8:7$ respectively. A man riding a bicycle completes one lap of this field along its perimeter at the speed of $28.8\text{km/hr}$ in $2.5\text{ minutes}$. What is the area of the field?

Step 1: Calculate the perimeter of the field.
The distance covered in one lap along the boundary is equal to the perimeter of the rectangular field.

* Speed, $v=28.8\text{km/hr}=28.8\times \frac{5}{18}\text{m/s}=1.6\times 5\text{m/s}=8\text{m/s}$
* Time, $t=2.5\text{ minutes}=2.5\times 60\text{ seconds}=150\text{ seconds}$

$$\text{Perimeter}=\text{Speed}\times \text{Time}=8\text{m/s}\times 150\text{ s}=1200\text{m}$$

Step 2: Find the dimensions of the rectangular field.
Let length $l=8x$ and breadth $b=7x$.

$$\text{Perimeter}=2(l+b)$$

$$1200=2(8x+7x)$$

$$1200=2(15x)⟹30x=1200⟹x=40$$

So, the dimensions are:

* Length, $l=8\times 40=320\text{m}$
* Breadth, $b=7\times 40=280\text{m}$

Step 3: Calculate the area of the field.

$$\text{Area}=l\times b=320\times 280=89600\text{ sq. m}$$