A path 5 metre wide runs along inside a rectangualar field. The length of the rectangular field is three times the breadth of the field. If the area of the path is 500 square metre,then find the length and breadth of the field
Answers
Step-by-step explanation:
Let the breadth of the rectangular field be x m
Length of the rectangular field = 3 times the breadth = 3 * x = 3x m
Area of the rectangular field = Length * Breadth = 3x * x = 3x² m²
Given
A Path 5m wide runs along inside a rectangular field.
Therefore find the dimenisions and area of rectangular field when path is not included
Width of the path = 5 m
Length of the rectangular field when path is not included = 3x - 2(Width of the path) = 3x - 2(5) = 3x - 10 = (3x - 10) m
Breadth of the rectangular field when path is not included = x - 2(Width of the path) = x - 2(5) = x - 10 = (x - 10) m
Area of the rectangular field when path is not included = Length * Breadth
= (3x - 10)(x - 10)
= 3x(x - 10) - 10(x - 10)
= 3x² - 30x - 10x + 100
= 3x² - 40x + 100
Given
Area of the path = 500 m²
i.e Area of the rectangular field - Area of the rectangular field when path is not included = 500 m²
\implies 3 {x}^{2} - (3 {x}^{2} - 40x + 100) = 500⟹3x
2
−(3x
2
−40x+100)=500
\implies 3 {x}^{2} - 3 {x}^{2} + 40x - 100 = 500⟹3x
2
−3x
2
+40x−100=500
\implies 40x - 100 = 500⟹40x−100=500
\implies 40x - 100 = 500⟹40x−100=500
\implies 40x = 500 + 100⟹40x=500+100
\implies 40x = 600⟹40x=600
\implies 4x = 60⟹4x=60
\implies x = \dfrac{60}{4}⟹x=
4
60
\implies \boxed{x = 15}⟹
x=15
Breadth of the rectangle = x = 15 m
Length of the rectangle = 3x = 3 * 15 = 45 m
Hence, length and breadth of the field are 45 m and 15 m respectively.