In the centre of a rectangular lawn of dimensions 50 m × 40 m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 m2 . Find the length and breadth of the pond.
Answers
Dimensions = 50 m and 40 m
Area of the rectangular lawn = 50 × 40 = 2000 m²
Area of the grass surrounding the pond = 1184 m²
So, the area of the pond = area of the lawn - area of the grass
= 2000 - 1184
Area of the pond = 816 m²
Let the width of around the pond be 'x' m
Then, the length of the pond = (50 - 2x) m
and the breadth of the pond = (40 - 2x) m
Area of the pond = 816 m²
⇒ (50 - 2x) × (40 - 2x) = 816
2000 - 80x - 100x + 4x² = 816
4x² - 180x + 2000 - 816 = 0
4x² - 180x + 1184 = 0
dividing it by 4 we get
x² - 45x + 296 = 0
x² - 37x - 8x + 296 =0
x(x - 37) - 8(x - 37) = 0
(x - 37) (x - 8) = 0
x = 37 or x = 8
suppose if x = 37, then
the length of the pond will be 50 - 2*37
50 - 74 = - 24m, which is not possible because length cannot be negative.
Therefore the length of the pond = 50 - 2*8
50 - 16 = 34 m
and
Breadth of the pond = 40 - 2*8
40 - 16 = 24 m
Length = 34 m and breadth = 24 m
Answer:
Step-by-step explanation:
Given, dimensions of rectangular lawn = 50 m × 40 m
Now, Area of the rectangular lawn = 50 × 40 = 2000 m2
Area of the grass surrounding the pond = 1184 m2
Therefore, area of the rectangular pond = Area of the rectangular lawn - Area of the grass surrounding the pond = 2000 - 1184 = 816 m2
Let the width of the surrounding grass be x.
Therefore, length of rectangular pond = 50 - 2x
and breadth of rectangular pond = 40 - 2x
Again, area of rectangular pond = 816
⇒ (50 - 2x)×(40 - 2x) = 816
⇒ 2000 - 80x - 100x + 4x2 = 816
⇒ 4x2 - 180x + 2000 - 816 = 0
⇒ 4x2 - 180x + 1184 = 0
⇒ x2 - 45x + 296 = 0
⇒ x2 - 37x - 8x + 296 = 0
⇒ x(x-37) - 8(x-37)= 0
⇒ (x-8)(x-37)= 0
⇒ x = 8 or 37
Now, x ≠ 37, because for x = 37,
length of rectangular pond = 50 - 2(37) = -24 m which isn't possible.
Therefore, length of rectangular pond = 50 - 2(8) = 34 m
and breadth of rectangular pond = 40 - 2(8) = 24 m