The length of a rectangular field is 34 m more than its breadth. Its area is 5040 m². Its breadth (in m) is:

The correct option is 56.

Let the breadth of the rectangular field be represented by $b$ meters.
According to the problem, the length of the rectangular field is 34 m more than its breadth. Therefore, the length ($l$) can be expressed as:

$l=b+34$

The area of a rectangle is calculated by multiplying its length and breadth ($\text{Area}=l\times b$). We are given that the area is 5040 m². Substituting the expression for length, we get:

$b(b+34)=5040$

Expanding the equation gives a quadratic equation in terms of $b$:

$b^2+34b-5040=0$

To solve this quadratic equation, we can factor it by finding two numbers that multiply to -5040 and add up to 34. These two numbers are 90 and -56. Splitting the middle term, we write:

$b^2+90b-56b-5040=0$

Now, factor by grouping:

$b(b+90)-56(b+90)=0$

$(b-56)(b+90)=0$

This gives two potential values for $b$:

$b=56\text{or}b=-90$

Since the breadth of a physical field cannot be negative, we discard $b=-90$.
Thus, the breadth of the rectangular field is 56 m.