The rate of change of the area of a square with respect to the length of its diagonal, when the side of the square is 4 cm, is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(4\sqrt{2}\) cm
Let l be the length of the diagonal and A the area of the square. The side is \(\frac{l}{\sqrt{2}}\), so the area in terms of l is
\( \displaystyle A = \left(\frac{l}{\sqrt{2}}\right)^{2} = \frac{l^{2}}{2} \)
Differentiating with respect to l:
\( \displaystyle \frac{dA}{dl} = l \)
When the side is 4 cm, the diagonal is \(4\sqrt{2}\) cm, so the rate is \(4\sqrt{2}\) cm (cm2 of area per cm of diagonal).
Why the other options are incorrect
Option (1) 8 cm — Incorrect. With side x, \(\frac{dA}{dx} = 2x = 8\) at x = 4: this is the rate with respect to the side, not the diagonal.
Option (3) \(2\sqrt{2}\) cm — Incorrect. This is half the diagonal; since \(A = \frac{l^{2}}{2}\), the derivative is l itself.
Option (4) 16 cm — Incorrect. This is the area 16 cm2 of the square, not a rate of change.