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 = x2cm2
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))
\( \dfrac{dA}{dt}\)= 64cm2/min
Thus the area is increasing at the rate of 64cm2/min when the side is 8cm long.
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