Question

$\huge \bf Question$

There is a square field whose side is 44 m. A square flower bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and gravelling the path at ₹ 2.75 and ₹ 1.50 per square metre, respectively, is ₹4904. Find the width of the gravel path.

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Answer 1

Answer :-

Given :-

• Side of square field = 44 m
• Cost of laying flower bed = ₹ 2.75 per m²
• Cost of gravelling the path = ₹ 1.5 per m²
• Total cost = ₹ 4904

To Find :-

• Width of gravel path

Solution :-

Let the width of gravel path be x

As the flower bed is in the center,

Each side of flower bed = 44 - x - x = 44 - 2x m

Area of gravel path = Area of square field - Area of flower bed

Calculating Area of square field and flower bed :-

➣ Area of square field = 44 × 44 = 1936 m²

Area of flower bed = ( 44 - 2x ) ( 44 - 2x )

= 1936 - 88x - 88x + 4x²

➣ Area of flower bed = 4x² + 1936 - 176x

So, the area of gravel path =

= 1936 - 4x² - 1936 + 176x

➣ Area of gravel path = 176x - 4x²

Cost of gravelling path = Area × cost per m²

= ( 176x - 4x² ) × 1.5

= 1.5 × 176x - 1.5 × 4x²

➣ Cost of gravelling path = 264x - 6x²

Cost of Laying flower bed = Area × cost per m²

= ( 4x² + 1936 - 176x ) × 2.75

➣ Cost of Laying flower bed = 11x² + 5324 - 484x

According to the question :-

➱ 11x² + 5324 - 484x + 264x - 6x² = 4904

➱ 11x² - 6x² - 484x + 264x + 5324 = 4904

➱ 5x² - 220x + 5234 = 4904

➱ 5x² - 220x + 330 = 0

Dividing the equation by 5

➱ x² - 44x + 66 = 0

By using quadratic formula :-

$\sf x = \frac{-b\pm\sqrt{b^2 - 4ac }}{2a}$

➱ $\sf x = \frac{-(-44) \pm\sqrt{44^2 - 4 (66) }}{2 (1)}$

➱ $\sf x = \frac{44 \pm \sqrt{1672}}{2}$

➱ $\sf x = 42 , 2 \: (approx)$

Width of gravel path can't be 42 m as the length of square field is 44 m. So this value is not possible.

Hence, x = 2

Width of gravel path = 2 m

Answer 2

Let the width of the gravel path be x metres. Then, Each side of the square flower bed is (44−2x) metres

Now, Area of the square field 44×44=1936m2

Area of the flower bed =(44−2x)2m2

Area of the gravel path= Area of the field -  Area of the flower bed

                                      =1936−(44−2x)2=1936−

(1936−176x+4x²)=(176×+4ײ)m²

Cost of laying the flower bed=Area of the flower bed × Rate per sq.m

                                                 =(44−2x)2×100275=411(44−2x)2=11(22−x)2

Cost of gravelling the path = Area of the path × Rate per sq. m

                                            =(176x−4x2)100150=6(44−x2)

It is given that the total cost of laying the flower bed and gravelling the path is Rs. 4904

∴11(22−x)2+6(44x−x2)=4904

⇒5x2−220x+5324=4908

⇒5x2