An agricultural field is in the form of a rectangle having length X₁ metres and breadth X₂ metres (X₁ and X₂ are variable). If X₁ + X₂ = 40 metres, then the area of the agricultural field will not exceed which one of the following values ?

The correct option is 400 sq m.

Step-by-Step Explanation:

Let the length of the rectangular agricultural field be X₁ metres and the breadth be X₂ metres.
The area (A) of a rectangle is calculated as the product of its length and breadth:

$$A=X_1\times X_2$$

We are given the relation:

$$X_1+X_2=40$$

From this relation, we can express one variable in terms of the other:

$$X_2=40-X_1$$

Now, substitute this expression for X₂ into the formula for area:

$$A=X_1(40-X_1)$$

$$A=40X_1-X_1^2$$

To find the maximum possible area, we can rewrite this quadratic expression by completing the square:

$$A=-(X_1^2-40X_1)$$

Add and subtract 400 inside the parenthesis (since ${\left(\frac{40}{2}\right)}^2=400$):

$$A=-(X_1^2-40X_1+400-400)$$

$$A=400-{(X_1-20)}^2$$

Since the squared term ${(X_1-20)}^2$ must always be greater than or equal to 0 for any real value of X₁, the maximum value of A occurs when ${(X_1-20)}^2=0$ (which happens when $X_1=20$).
Thus, the area will always satisfy the inequality:

$$A\le 400$$

Therefore, the area of the agricultural field will not exceed 400 sq m.