Question

The side of a square sheet is increasing at the rate of 4 cm per minute. At what rate is the area increasing when the side is 8 cm long?

Answer 1

\[\text { Given }: A = x^2 \text { and } \frac{dx}{dt}= 4 cm/min \]
\[\text { Let  x be the side of the square and A be its area at any time t.Then },\]
\[A = x^2 \]
\[ \Rightarrow \frac{dA}{dt} = 2x\frac{dx}{dt}\]
\[ \Rightarrow \frac{dA}{dt} = 2 \times 8 \times 4 \left[ \because x = 8 cm\text { and } \frac{dx}{dt} = 4 cm/\min \right]\]
\[ \Rightarrow \frac{dA}{dt} = 64 \ {cm}^2 /\min\]