The volume of a cube with sides of length #s# is given by #V =s^3#. What is the rate of change of the volume with respect to #s# when # s# is 6 centimeters?

Answer 1

#color(blue)(108cm^3)#

This is the volume function that we have:

#V=s^3#

The first derivative of this function, which is also known as the gradient function because it provides the gradient of a tangent line drawn at the given point, is used to find the rate of change.

#(dV)/(ds)(s^3)=3s^2#

We now plug in #s=6#

#3s^2=3(6)^2=108(cm^3)/s#

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Answer 2

Theodore Abad

To find the rate of change of the volume ((V)) with respect to the side length ((s)) when (s) is 6 centimeters, we take the derivative of the volume formula (V = s^3) with respect to (s). Then, we substitute (s = 6) into the derivative.

The derivative of (V) with respect to (s) is: [ \frac{dV}{ds} = 3s^2 ]

Substituting (s = 6): [ \frac{dV}{ds} = 3(6)^2 ] [ \frac{dV}{ds} = 3(36) ] [ \frac{dV}{ds} = 108 ]

Therefore, when the side length (s) is 6 centimeters, the rate of change of the volume with respect to (s) is 108 cubic centimeters per centimeter.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.