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Compute the rate of change of volume of a cube given the rate of change of an edge

Given a cube with edges expanding at a rate of 1 centimeter per second, what is the rate of change of the volume of the cube when the length of an edge is:

  1. 5cm.
  2. 10cm.
  3. x cm.

First, we know the volume of a cube is given by

    \[ V = e^3 \]

where e is the length of an edge. Thus, the rate of change of the volume relative to the rate of change of an edge is given by

    \[ \frac{dV}{de} = 3e^2. \]

We use this formula to find the rate of change of the volume for given values of e,

  1. If e = 5 then we have

        \[ \frac{dV}{de} = 3 (5)^2 = 75 \text{ cm}^3 / \text{ sec}. \]

  2. If e = 10 then we have

        \[ \frac{dV}{de} = 3(10)^2 = 300 \text{ cm}^3 / \text{ sec}. \]

  3. If e = x then we have

        \[ \frac{dV}{de} = 3 x^2 \text{ cm}^3 / \text{ sec}. \]

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