Question

The perimeter of a rectangular field is 2.5Kilometer. What is the greatest possible area it may contain?

Answer 1

Hint: Here we use the concept that a square is also a rectangle with length and breadth equal. A rectangle will have one of the measurements less than the other, so it will have the greatest area if both the measurements are equal. Substitute the given perimeter to the perimeter of the square and find the value of the side of the square. Using the formula of area of square we find the greatest area.
* Perimeter of a square with side a is \[4a\]
* Area of a square having length of side a is \[{a^2}\]

Complete step-by-step answer:
We are given the perimeter of a rectangle as 2.5Km
We know a rectangle has a length ‘\[l\]’ and a breadth ‘\[b\]’.
Then the area of a rectangle is \[l \times b\].
We know in a rectangle either length is greater or breadth is greater in measurement.
If we take length and breadth equal in measurement (say\[a\]), then the rectangle becomes a square as all sides become equal in measurement. So a rectangle can have the greatest area if its two sides are equal in measurement( else one side will be less than the other side).
If we take perimeter of a square \[ = 2.5\]Km
From the formula of perimeter of a square, we can write
\[ \Rightarrow 4 \times a = 2.5\]
Divide both sides by 4
\[ \Rightarrow \dfrac{{4 \times a}}{4} = \dfrac{{2.5}}{4}\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow a = 0.625\]Km
We know squares with length '\[a\]’ have area\[{a^2}\].
Area of square \[ = {(0.625)^2}\]
Calculating the value we get
Area of square \[ = 0.390625\]\[K{m^2}\]
Therefore, the greatest possible area a field of perimeter 2.5Km can contain is \[0.390625\]\[K{m^2}\].

Note: Students make mistakes forming the equation of perimeter using length and breadth and then substituting in the formula for area of a rectangle, this will yield nothing as we will get an abrupt quadratic equation which we will not be able to solve. We have to find the greatest area the field can have so we look at the possibility of sides of the rectangle being maximum.