Of all the rectangles of area 49 cm2, the one with the least perimeter has a perimeter of
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 28 cm
Let the sides be x and y cm, with xy = 49, so \(y = \displaystyle\frac{49}{x}\). The perimeter is
\( \displaystyle P = 2\left(x + \frac{49}{x}\right),\ x \gt 0 \)
Setting the derivative equal to zero:
\( \displaystyle \frac{dP}{dx} = 2\left(1 - \frac{49}{x^2}\right) = 0 \)
So \(x^2 = 49\), and x = 7 (x > 0).
\( \displaystyle \frac{d^2P}{dx^2} = \frac{196}{x^3} \gt 0 \)
So P is least at x = 7, where y = 7 (a square), and
\(P = 2(7 + 7) = 28\) cm
Why the other options are incorrect
Option (1) 14 cm — Incorrect. This is x + y = 14, half the perimeter.
Option (2) 24 cm — Incorrect. No rectangle of area 49 cm2 has a perimeter below 28 cm.
Option (4) 56 cm — Incorrect. This doubles the least perimeter; it is not the least value.