Question

The diagonal of a square is changing at the rate of $\frac{1}{2}\mathrm{cm}/\sec .$ Then the rate of change of area, when the area is 400 cm², is equal to ____________________.

Answer 1

Let the side of the square be x cm at any time t.

∴ Length of the diagonal of square, l = $\sqrt{2}$ × Side of square = $\sqrt{2}x$

$l=\sqrt{2}x$

Differentiating both sides with respect to t, we get

$\frac{dl}{dt}=\sqrt{2}\times \frac{dx}{dt}$

It is given that, $\frac{dl}{dt}=\frac{1}{2}$ cm/sec

$\therefore \frac{1}{2}=\sqrt{2}\times \frac{dx}{dt}$

$\Rightarrow \frac{dx}{dt}=\frac{1}{2\sqrt{2}}$ cm/sec

Now,

Area of the square, A = x²

A = x²

Differentiating both sides with respect to t, we get

$\frac{dA}{dt}=2x\frac{dx}{dt}$ .....(1)

When A = 400 cm²,

x² = 400 cm² = (20 cm)²

⇒ x = 20 cm

Putting x = 20 cm and $\frac{dx}{dt}=\frac{1}{2\sqrt{2}}$ cm/sec in (1), we get

$\frac{dA}{dt}=2\times 20\times \frac{1}{2\sqrt{2}}=10\sqrt{2}$ cm²/sec

Thus, the rate of change of area of the square is $10\sqrt{2}$ cm²/sec.


The diagonal of a square is changing at the rate of $\frac{1}{2}\mathrm{cm}/\sec .$ Then the rate of change of area, when the area is 400 cm², is equal to $\underline{10\sqrt{2}{\mathrm{cm}}^2/\sec }$.