Question

$\huge \bf {Question}$
Area of a rectangle is 325 cm².The perimeter of rectangle is 76 cm. Find its Dimensions

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Answer 1

GIVEN :

• Area of rectangle = 325cm²
• Perimeter of rectangle = 76cm

To find :

Dimensions of rectangle

Formula used :

• Area of rectangle = length × breadth
• Perimeter of rectangle = 2×( length + Breadth)

Solution :

Let -

• Length = x cm
• Breadth = y cm

Therefore perimeter of rectangle = 2(x+y)

This gives,

➝ 2(x+y) = 76

➝ x+y = 76÷2

➝ x+y = 38

➝ y = 38 - x ....... equation 1

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Also , area = xy

Therefore

➝ xy = 325

On putting value of y from equation 1 into above equation, we get;

➝ x(38-x) = 325

➝ 38x - x² = 325

➝ x² - 38x + 325 = 0

Now , the above formed Equation is a quadratic equations, therefore using quadratic formula,

$\sf x \: = \dfrac{( - b) \: \pm \: \sqrt{ {b}^{2} - 4ac} }{2a} \\ \\ \sf \implies \: x \: = \dfrac{ - ( - 38) \: \pm \: \sqrt{ {( - 38)}^{2} \: - \: 4(1)(325) } }{2(1)} \\ \\ \sf \implies \: x \: = \: \dfrac{38 \: \pm \: \sqrt{1444 \: - \: 1300} }{2} \\ \\ \sf \implies \: x \: = \: \dfrac{38 \: \pm \: \sqrt{144} }{2} \\ \\ \sf \implies \: x \: = \: \dfrac{38 \: \pm \: 12}{2} \\ \\ \sf \implies \: x \: = \: \dfrac{2(19 \: \pm \: 6)}{2} \\ \\ \sf \implies \: x \: = \: 19 \: \pm \: 6$

Therefore,

either , x = 19 + 6 .......or........ x = 19 - 6

either , x = 25 .........or........... x = 13

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When , x = 25

y = 38 - 25

y = 13

when x = 13

y = 38 - 13

y = 25

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ANSWER :

Dimensions of required rectangle are , 13cm × 25cm

Answer 2

Given :-

• Area of a rectangle is 325 cm²

• The perimeter of rectangle is 76 cm.

To find :-

• The dimensions of the rectangle

Solution :-

Let L cm be the length of the rectangle and B cm be the breadth of the rectangle.

First, let's get the relation between length and breadth of the rectangle.

Perimeter of the rectangle = 2 (length + breadth)

➙ Perimeter of the rectangle = 2 (L + B)

➙ 76 cm = 2(L + B)

➙ L + B = 76/2

➙ L + B = 38 cm

➙ L = (38 - B) cm

• Now, substitute the value of length in area of rectangle formula. Then we'll get a quadratic equation. By solving the equation, we get the dimensions of the rectangle.

Area of the rectangle = length × breadth

➙ Area of the rectangle = L × B

➙ 325 cm² = (38 - B) (B)

➙ 325 = 38B - B²

➙ B² - 38B + 325 = 0

We'll solve this equation by factorization.

➙ B² - 25B - 13B + 325 = 0

➙ B(B - 25) - 13(B - 25) = 0

➙ (B - 25) (B - 13) = 0

➛ B - 25 = 0 ; B = 25

➛ B - 13 = 0 ; B = 13

So, the breadth of the rectangle is either 25 cm or 13 cm

Let B = 13 cm

L + B = 38

L + 13 = 38

L = 38 - 13

L = 25 cm

Therefore,

• The length of the rectangle = 25 cm

• The breadth of the rectangle = 13 cm