The side of a square is increasing at the rate of 4 cm/s. The rate at which the area of the square is increasing when the side is 7 cm long is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 56 cm2/s
Let x be the side and A the area at time t, so \(A = x^2\). By the chain rule:
\( \displaystyle \frac{dA}{dt} = 2x \frac{dx}{dt} \)
With x = 7 cm and \(\frac{dx}{dt} = 4\) cm/s:
\( \displaystyle \frac{dA}{dt} = 2 \times 7 \times 4 \)
\( = 56 \) cm2/s
Why the other options are incorrect
Option (1) 28 cm2/s — Incorrect. This is \(7 \times 4\); the factor 2 in the derivative \(2x\) of \(x^2\) is missing.
Option (3) 16 cm2/s — Incorrect. This is 4 × 4, the rate of increase of the perimeter 4x (in cm/s), not of the area.
Option (4) 196 cm2/s — Incorrect. This is \(4x^2\), the area 49 multiplied by the rate 4 instead of differentiating.