# How do you find the instantaneous rate of change for the volume of a growing spherical cell given by #v = (4/3) (pi) r^3# when r is 5?

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To find the instantaneous rate of change for the volume of a growing spherical cell at ( r = 5 ), you need to take the derivative of the volume function ( v = \frac{4}{3} \pi r^3 ) with respect to ( r ) and then evaluate it at ( r = 5 ). The derivative of ( v ) with respect to ( r ) is ( \frac{dv}{dr} = 4 \pi r^2 ). Evaluating this at ( r = 5 ), we get ( \frac{dv}{dr} = 4 \pi (5)^2 = 100 \pi ) cubic units per unit length. Therefore, the instantaneous rate of change for the volume of the spherical cell when ( r = 5 ) is ( 100 \pi ) cubic units per unit length.

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