Page 69 - Mathematics Class - XII

DEMONSTRATION
1. Area of rectangle, A = 15 cm × 7 cm = 105 cm 2
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Area of rectangle, A = 14 cm × 8 cm = 112 cm 2
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Area of rectangle, A = 13 cm × 9 cm = 117cm 2
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Area of rectangle, A = 12.5 cm × 9.5 cm = 118.75 cm 2
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Area of rectangle, A = 12 cm × 10 cm = 120 cm 2
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Area of rectangle, A = 11 cm × 11 cm = 121 cm 2
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Area of rectangle, A = 10.5 cm × 11.5 cm = 120.75 cm 2
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2. Perimeter of each rectangle is same but their areas are different. Area of rectangle A is the maximum.
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It is a square of side 11 cm.

OBSERVATION
1. Perimeter of each rectangle A , A , A , A , A , A , A is 44 cm.
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2. Area of the rectangle A is less than the area of rectangle A .
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3. Area of the rectangle A greater than the area of rectangle A .
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4. The rectangle A has the dimensions 11 × 11 and hence it is a square.
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5. Of all the rectangles with same perimeter, the rectangle A has the maximum area.
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CONCLUSION
From this activity it is verified that amongst all the rectangles of the same perimeter, the square has the
maximum area.

APPLICATION
This activity is useful in explaining the idea of finding maximum of a function.

Knowledge Booster

Let the length and breadth of rectangle be x and y.
The perimeter of the rectangle A = 44 cm.
2 (x + y) = 44
or x + y = 22
or y = 22 – x
Let A (x) be the area of rectangle, then
A (x) = xy
= x (22 – x)
= 22x – x 2
A′ (x) = 22 – 2x
A′ (x) = 0 ⇒ 22 – 2x = 0 ⇒ x = 11
A˝ (x) = – 2
A˝ (11) = – 2, which is negative
Therefore, area is maximum when x = 11
y = 22 – x = 22 – 11 = 11
So, x = y = 11
Hence, amongst all rectangles, the square has the maximum area.

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