Question

if the perimeter of the rectangle is 82 metres and the area is 400 m square the breadth of the rectangle is

Answer 1

Answer :

We know that,

perimeter of a rectangle = 2 × (length + breadth)

and

area of a rectangle = length × breadth

Let, length = a m and breadth = b m

By the given conditions,

perimeter of the rectangle = 82 m

⇒ 2(a + b) = 82

⇒ a + b = 41 ...(i)

and

area of the rectangle = 400 m²

⇒ ab = 400 ...(ii)

From (i), we get

a = 41 - b

Substituting a = 41 - b in (ii), we get

(41 - b)b = 400

⇒ 41b - b² = 400

⇒ b² - 41b + 400 = 0

⇒ b² - (25 + 16)b + 400 = 0

⇒ b² - 25b - 16b + 400 = 0

⇒ b(b - 25) - 16(b - 25) = 0

⇒ (b - 25)(b - 16) = 0

∴ b = 25, b = 16

So, the breadth of the given rectangle is either 25 m or 16 m.

[N.B. - It is better to write down the breadth of the rectangle as 16 m and to consider the length as 25 m]

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Answer 2

SOLUTION :  

Given :  

Perimeter of a rectangular field = 82 m

Area of a rectangular field = 400 m²

Let the breadth of a rectangle be 'b’ m.

Perimeter of a rectangle = 2(l + b)

82 = 2(l + b)

82/2 = (l + b)

l + b = 41  

Length,l = 41 - b …………(1)

Area of a rectangle  = l × b

400 = (41 - b) b

[From eq 1]

400 = 41b - b²

b² - 41b + 400 = 0

b² - 25b - 16b + 400 = 0

[By middle term splitting]

b(b - 25) - 16(b - 25) = 0

(b - 25) (b - 16) = 0

(b - 25) = 0  or  (b - 16) = 0

b = 25 or b = 16  

If breadth,b = 25 , then length, l = 41 - b  

l = 41 - 25 = 16 m

If breadth,b = 16 , then length, l = 41 - b  

l = 41 - 16 = 25 m

Since, length is more than breadth  

Therefore , breadth, b = 16 m

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