Question

3. A farmer wishes to start a 100 sq.m rectangular vegetable garden. Since he has only
30m barbed wire, he fences the sides of the rectangular garden letting his house
compound wall act as the fourth side fence. Find the dimension of the garden.

Answer 1

Let the length of rectangular garden is l and breadth is b.
A/c to question, Garden is fenced on three sides.
∴ l + 2b = 30
l = 30 - 2b ------(1)

Given, area of rectangular garden = 100 m²
l × b = 100
(30 - 2b)b = 100
30b - 2b² = 100
2b² - 30b + 100 = 0
b² - 15b + 50 = 0
b² - 10b - 5b + 50 = 0
(b - 10)(b - 5) = 0
b = 5 and 10
So, l = 30 - 2b ⇒l = 20 or 10

Hence, rectangular garden sides (20, 5) or (10, 10)

Answer 2

Let ‘x’ and ‘y’ be the length & breadth of the vegetable garden

GIVEN : Area of vegetable garden = 100 m²
Area of rectangle = Length x breadth
x x y = 100
x = 100/y…………….(1)

For fencing We calculate the perimeter.
Here, the fencing will be on three sides of the vegetable garden because one side of the vegetable garden will act as the compound wall.

Given - Perimeter of 3 sides of the vegetable garden= 30 m
x + x + y = 30
2 x + y = 30
2(100/y) + y = 30
[From eq 1]
(200/y) + y = 30
(200 + y²)/y = 30
200 + y² = 30 y
y² - 30 y + 200 = 0
y² - 10 y – 20 y + 200 = 0
[By middle term splitting
y (y – 10) - 20 (y – 10) = 0
(y – 10) (y – 20) = 0
y – 10 = 0 or y – 20 = 0
y = 10 or y = 20

Put these values of y in eq 1,
x = 100/y

If y = 10
x =100/10
x = 10 m

If y = 20
x = 100/20
x = 5 m

Hence, the required dimensions are (10 m ,10 m) or (20 m , 5 m).

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