Given the side of a square sheet is increasing at the rate of 4 cm per minute.

To find rate of area increasing when the side is 8 cm long

Let the side of the given square sheet be x cm at any instant time.

Then according to the given question, we can write as

The rate of side of the sheet increasing is, \( \dfrac{dx}{dt}\) = 4cm/min ...(i)

Now the area of the square sheet at any time t will be A = x^{2}cm^{2}

By applying derivative with respect to time on both sides

\( \dfrac{dA}{dt}\) = \( 2x\times 4 = 8x\)............(ii)

From the equation (i)

Therefore when the side is 8cm long,

the rate of area increasing will become \( \dfrac{dA}{dt}\) = 8 x 8 (from the equation (ii))

\( \dfrac{dA}{dt}\)= 64cm^{2}/min

Thus the area is increasing at the rate of 64cm^{2}/min when the side is 8cm long.

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