What is the length of a rectangular garden whose perimeter is 32cm and area 39cm2?

Analyze the question: We need the length of a rectangle with perimeter 32 cm and area 39 cm².

Concept:

• Perimeter: 2(l + w) = 32, so l + w = 16
• Area: l × w = 39

Solution:

From l + w = 16, we get w = 16 − l.

Substitute into the area equation: l(16 − l) = 39 16l − l² = 39 l² − 16l + 39 = 0

Factorise: we need two numbers that multiply to 39 and add to 16. These are 13 and 3. (l − 13)(l − 3) = 0 l = 13 or l = 3

If l = 13, then w = 3 (the length is the longer side). If l = 3, then w = 13, which would make the "length" shorter than the width. So the length is 13 cm.

Check:

• Perimeter: 2(13 + 3) = 32 ✓
• Area: 13 × 3 = 39 ✓

Option D (9 cm) would give a width of 7 cm and an area of 63 cm², so it doesn't fit. Option A (25 cm) is bigger than half the perimeter, so it is impossible. Option B (18 cm) is also more than 16, so it is impossible.

Exam tip: Whenever you have perimeter and area of a rectangle, find l + w = P/2 first. Then look for two numbers whose sum is P/2 and product is the area. You can also test the options quickly: the length must be less than 16 here, which eliminates A and B immediately.