The length of a rectangle is always twice its width. The width is increasing at the rate of 3 cm/s. The rate at which the area of the rectangle is increasing when the width is 5 cm is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 60 cm2/s
Let w be the width at time t. Then the length is 2w, and the area is
\(A = 2w \times w = 2w^2\)
Differentiating with respect to t by the chain rule:
\( \displaystyle \frac{dA}{dt} = 4w\,\frac{dw}{dt} \)
\( = 4 \times 5 \times 3 \)
\( = 60 \) cm2/s
Why the other options are incorrect
Option (1) 30 cm2/s — Incorrect. This is \(2w \times 3\), the rate for a square of side w; the length 2w is not used.
Option (3) 50 cm2/s — Incorrect. This is the area \(2 \times 5^2\) itself, not its rate of change.
Option (4) 20 cm2/s — Incorrect. This is dA/dw = 4w at w = 5, leaving out the rate 3 cm/s of the width.