a field is 70m long and 40m broad. a path of uniform width runs all around inside the field and covers an area of 1000m².find the width of the path
Answers
Answer:
The width of the path is 5 m.
Step-by-step explanation:
Given :
- Length of the field = 70 m
- Breadth of the field = 40 m
- Area of the path = 1000 m²
To find :
the width of the path.
Solution :
Area of the field (outer rectangle) = length × breadth of the field
⇒ Area of the field = 70 × 40
⇒ Area of the field = 2800 m²
Let 'x' m be the width of the path.
Dimensions of the inner rectangle :
- Length = (70 - 2x) m
- Breadth = (40 - 2x) m
Area of the inner rectangle = (70 - 2x) (40 - 2x)
⇒ Area of the inner rectangle = 70(40 - 2x) - 2x(40 - 2x)
⇒ Area of the inner rectangle = 2800 - 140x - 80x + 4x²
⇒ Area of the inner rectangle = (4x² - 220x + 2800) m²
Area of the path = Area of the field(outer rectangle) - Area of the inner rectangle
⇒ 2800 - (4x² - 220x + 2800) = 1000
⇒ 2800 - 4x² + 220x - 2800 = 1000
⇒ -4x² + 220x = 1000
⇒ 4x² - 220x + 1000 = 0
⇒ 4(x² - 55x + 250) = 0
⇒ x² - 55x + 250 = 0
⇒ x² - 50x - 5x + 250 = 0
⇒ x(x - 50) - 5(x - 50) = 0
⇒ (x - 50) (x - 5) = 0
⇒ x = 50 m , 5 m
x can not be 50 m. So, x = 5 m
Therefore, the width of the path is 5 m.
