Question

The perimeter of a rectangular field is 82 m and it's area is 400 m². Find the dimension of the field.

Answer 1

Answer:

We know that 2(length + breadth) = perimeter.

$\sf \therefore \: (length + breadth)$

$= \sf \: \frac{1}{2} \times perimeter$

$\sf = \frac{1}{2} \times 82 \: m$

$\sf = 41 \: m$

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Let the length of the field be x meters.

Then, it's breadth= (41 - x) m.

$\sf \therefore area \: of \: the \: field = x(41 - x) {m}^{2}$

$\sf = (41x - {x}^{2} ) {m}^{2}$

But, area = 400 m² (given)

$\sf \therefore \: 41x - {x}^{2} = 400$

$\sf \dashrightarrow \: {x}^{2} - 41x + 400 = 0$

$\sf \dashrightarrow \: {x}^{2} - 25x - 16x + 400 = 0$

$\sf \dashrightarrow \:x(x - 25) - 16(x - 25) = 0$

$\sf \dashrightarrow \:(x - 25)(x - 16) = 0$

$\sf \dashrightarrow \:x - 25 = 0 \: \: or \: \: x - 16 = 0$

$\sf \dashrightarrow \:x = 25 \: \: or \: \: x = 16$

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$\sf \therefore \: length \: of \: the \: field \: is \: \bold{25 \: m}.$

$\sf \therefore \: breadth \: of \: the \: field \: is \: \bold{ 16 \: m}.$

Answer 2

: REQUIRED ANSWER

$\implies$ let breadth be B

$\implies$ perimeter 2(L+b)

$\implies$ but perimeter = 82 m

$\implies$ Therefore, 2 (L+b) = 82

$\implies$ L+b = 41

$\implies$ therefore, L= 41 - b

$\implies$ Area = l/b

$\implies$ but Area Given = 400 m2

$\implies$ So L*b = 400

$\implies$ B* (41-B) = 400

$\implies$ -B2 + 41b = 400

$\implies$ - B2 + 41 B = 400= 0

$\implies$ B2 -41b+400=0 (Middle term factories)

$\implies$ B2 -(25b+16b)+400=0

$\implies$ B2- 25b-16b+400=0

$\implies$ B(B-25) - 16(B-25 )=0

Therefore, breadth and length are either 25 and 16 Respectively