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Question
An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 10 cm long?
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Solution
\[\text { Let x be the side and V be the volume of the cube at any time t. Then },\]
\[V = x^3 \]
\[ \Rightarrow \frac{dV}{dt} = 3 x^2 \frac{dx}{dt}\]
\[ \Rightarrow \frac{dV}{dt} = 3 \times \left( 10 \right)^2 \times 3 \left[ \because x=10 \ cm \ \text { and }\frac{dx}{dt}=3 cm/sec \right]\]
\[ \Rightarrow \frac{dV}{dt} = 900 {cm}^3 /\sec\]
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