A rectangle keeps a constant area of 48 cm2 while its length increases at the rate of 2 cm/s. The rate at which its width is decreasing when the length is 8 cm is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1.5 cm/s
Let x be the length and y the width at time t. Since the area is constant,
\(xy = 48\)
When x = 8 cm, y = 6 cm. Differentiating with respect to t by the product rule:
\( \displaystyle x\frac{dy}{dt} + y\frac{dx}{dt} = 0 \)
\( \displaystyle 8\frac{dy}{dt} + 6 \times 2 = 0 \)
\( \displaystyle \frac{dy}{dt} = -\frac{12}{8} = -1.5 \)
The minus sign means the width is decreasing, at 1.5 cm/s.
Why the other options are incorrect
Option (2) 0.75 cm/s — Incorrect. This is y/x = 6/8, leaving out the rate 2 cm/s of the length.
Option (3) 3 cm/s — Incorrect. The width falls at 1.5 cm/s; 3 cm/s is twice that.
Option (4) 6 cm/s — Incorrect. This is the width itself at that instant, not its rate.