Question

The adjacent sides of a rectangle with a given perimeter as \[100cm\] and enclosing maximum area are
A.\[10cm\] and \[40cm\]
B.\[20cm\] and \[30cm\]
C.\[25cm\] and \[25cm\]
D.\[15cm\] and \[35cm\]

Answer 1

Hint: Area and perimeter, in Maths, are the two important properties of two-dimensional figures.
Perimeter: Perimeter of a shape is defined as the total distance around the shape. Basically, it's the length of any shape if it is expanded in a linear form. The perimeter of different shapes can match in length with each other depending upon their dimensions.
Area: Area is the region bounded by the shape of an object. The space covered by the figure or any geometric shapes is the area of the shape. The area of all the shapes depends upon its dimensions and properties. Different shapes have different areas.

Complete step-by-step answer:
A rectangle is a parallelogram with four right angles. All rectangles are also parallelograms, but not all parallelograms are rectangles.
The perimeter \[P\] of a rectangle is given by the formula \[P = 2(l + b)\] where \[l\] is the length of the rectangle and \[b\] is the breadth of the rectangle.
The area \[A\] of a rectangle is given by the formula \[A = lb\] where \[l\] is the length of the rectangle and \[b\] is the breadth of the rectangle.
We are given that the perimeter of the rectangle is \[100cm\].
Which means \[2(l + b) = 100\]
Hence we get ,
\[l + b = 50\]
Therefore we get \[l = 50 - b\]
Now consider the area of rectangle \[A = lb\]
Therefore \[A = \left( {50 - b} \right)b\]
Hence we get ,
\[A = 50b - {b^2}\]
Now differentiating both side with respect to \[b\] we get
\[\dfrac{d}{{dx}}A = 50 - 2b\]
For maximum area , put \[\dfrac{d}{{dx}}A = 0\]
Therefore we get \[50 - 2b = 0\]
And hence \[b = 25\]
Therefore we get \[b = 25cm\]
Now substituting this value of \[b\] in value of \[l\] we get \[l = 50 - b\]\[ = 50 - 25 = 25\]
Therefore we get \[l = 25cm\]
Therefore option (C) is the correct answer.
So, the correct answer is “Option C”.

Note: Perimeter of a shape is defined as the total distance around the shape. Area is the region bounded by the shape of an object. All rectangles are also parallelograms, but not all parallelograms are rectangles.