Question

A rectangular garden 10 m by 16 m is to be surrounded by a concrete path of uniform width. Given that the area of the path is 120 sq.m. Assuming the width of the path to be x. Then find the new
length and width of the rectangular garden.
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Answer 1

Answer:

Length of the rectangular garden = 16 m

Breadth of the rectangular garden = 10 m

Area of the rectangular garden = 16*10 

= 160 m²

Area of the concrete path of width x meters (given) = 120 m²

Total area = area of rectangular garden + area of the path

⇒ 160 + 120 

= 280 m²

The width of the concrete path is uniform (Given)

So, 

Length of the whole rectangle (including concrete path) = (16 + 2x) m

Breadth of the whole rectangle (including concrete path) = (10 + 2x) m

⇒ (16 + 2x)*(10 + 2x) = 280

⇒ 4x² + 52x + 160 = 280

⇒ 4x² + 52x + 160 - 280 = 0

⇒ 4x² +  52x - 120 = 0

⇒ Dividing the whole equation by 4, we get.

⇒ x² + 13x - 30 = 0

⇒ x² + 15x - 2x - 30 = 0

⇒ x(x + 15) - 2(x + 15) = 0

⇒ (x - 2) (x + 15) = 0

⇒ x = 2 or x = - 15

As the width cannot be negative. So, the value of x is 2

x = 2 m

Length of the outer rectangle = 16 + (2*2) = 20 m

Breadth of the outer rectangle = 10 + (2*2) = 14 m

Answer 2

Answer:

Length of the rectangular garden = 16 m

Breadth of the rectangular garden = 10 m

Area of the rectangular garden = 16*10 

= 160 m²

Area of the concrete path of width x meters (given) = 120 m²

Total area = area of rectangular garden + area of the path

⇒ 160 + 120 

= 280 m²

The width of the concrete path is uniform (Given)

So, 

Length of the whole rectangle (including concrete path) = (16 + 2x) m

Breadth of the whole rectangle (including concrete path) = (10 + 2x) m

⇒ (16 + 2x)*(10 + 2x) = 280

⇒ 4x² + 52x + 160 = 280

⇒ 4x² + 52x + 160 - 280 = 0

⇒ 4x² +  52x - 120 = 0

⇒ Dividing the whole equation by 4, we get.

⇒ x² + 13x - 30 = 0

⇒ x² + 15x - 2x - 30 = 0

⇒ x(x + 15) - 2(x + 15) = 0

⇒ (x - 2) (x + 15) = 0

⇒ x = 2 or x = - 15

As the width cannot be negative. So, the value of x is 2

x = 2 m

Length of the outer rectangle = 16 + (2*2) = 20 m

Breadth of the outer rectangle = 10 + (2*2) = 14 m

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