If the rate of change of the area of a square | vedclass

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(C) Let the side length of the square and the cube be $x$.For the square $S$,the area $A = x^2$.Given that the rate of change of the area is equal to

Vedclass · Accepted Answer

$6$

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(C) Let the side length of the square and the cube be $x$.For the square $S$,the area $A = x^2$.Given that the rate of change of the area is equal to its side length,we have $\frac{dA}{dt} = x$.Since $\frac{dA}{dt} = \frac{d}{dt}(x^2) = 2x \frac{dx}{dt}$,we equate the two expressions:$2x \frac{dx}{dt} = x$.Assuming $x 
eq 0$,we get $\frac{dx}{dt} = \frac{1}{2}$.For the cube $C$,the volume $V = x^3$.The rate of change of the volume is $\frac{dV}{dt} = 3x^2 \frac{dx}{dt}$.Given that the rate of change of the side of the cube is the same as that of the square,we use $\frac{dx}{dt} = \frac{1}{2}$.At the time when the side length $x = 2$ units:$\frac{dV}{dt} = 3(2)^2 	imes \frac{1}{2} = 3 	imes 4 	imes \frac{1}{2} = 6$ $units^3/sec$.

If the rate of change of the area of a square $S$ is equal to its side length,and the rate of change of the side of $S$ is the same as that of a cube $C$,then the rate of change of the volume of $C$,at the time when its side length is $2$ units,will be ............ $units^3/sec$.

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