A rectangular pool is such that its breadth is 4 metres less than its length and its area is 96 sq. metres. Find the length and breadth of the pool.
Step-by-step solution
To find: Length and breadth of the pool
Idea: Let the length be x. Area = length × breadth gives a quadratic equation. Factorise it by splitting the middle term, and reject the negative value because a length cannot be negative.
- Let length = x m. Then breadth = (x − 4) m.½ mark
- Area: x(x − 4) = 96, so x2 − 4x − 96 = 0.½ mark
- Split −4x: need two numbers with product −96 and sum −4 → −12 and 8.
x2 − 12x + 8x − 96 = x(x − 12) + 8(x − 12) = (x − 12)(x + 8) = 0.1 mark - So x − 12 = 0 or x + 8 = 0, i.e. x = 12 or x = −8. A length cannot be negative, so reject x = −8.½ mark
- Length = 12 m, breadth = 12 − 4 = 8 m.½ mark
Check: 12 × 8 = 96 m2 ✓ and 8 is 4 less than 12 ✓.
Answer to write in the exam
Let length = x m, breadth = (x − 4) m
x(x − 4) = 96
x2 − 4x − 96 = 0
x2 − 12x + 8x − 96 = 0
(x − 12)(x + 8) = 0
x = 12 or x = −8 (rejected, a length cannot be negative)
∴ Length = 12 m, breadth = 12 − 4 = 8 m
Common mistakes that cost marks
- Keeping x = −8 as an answer. A length must be positive.
- Writing the breadth as x + 4 instead of x − 4 (the breadth is less than the length).
- Choosing 12 and −8 with the wrong signs: (x + 12)(x − 8) gives +4x, not −4x.
How this can come in the exam
The length of a rectangle is 3 m more than its breadth and its area is 40 m2. Its breadth is
- 4 m
- 5 m
- 8 m
- 10 m
Show answer
(B) 5 m
b(b + 3) = 40 → b2 + 3b − 40 = (b + 8)(b − 5) = 0 → b = 5.
The product of two consecutive positive even numbers is 168. Find them.
Show answer
n(n + 2) = 168 → n2 + 2n − 168 = 0 → (n + 14)(n − 12) = 0 (1 mark). n = 12 (positive), so the numbers are 12 and 14 (1 mark).Try one yourself
A rectangular garden has breadth 5 m less than its length and area 84 m2. Find its length and breadth.
Show answer
x(x − 5) = 84 → x2 − 5x − 84 = (x − 12)(x + 7) = 0 → x = 12. Length 12 m, breadth 7 m.
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