The edge of a cube is increasing at a rate of .05 cm/s. In terms of the side of the cube, s, what is the rate of change of the volume of the cube?
V = s^3
The rate of change of volume is :
dV/dt = 3s^2 ds/dt = 0.15 s^2
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To find the rate of change of the volume of a cube, we need to use the formula for the volume of a cube. The volume of a cube is given by V = s^3, where s represents the length of each side of the cube.
Now, we can differentiate both sides of the volume equation with respect to time (t) to find the rate of change of the volume with respect to time.
dV/dt = d/dt (s^3)
Here, dV/dt represents the rate of change of volume, and d/dt represents the derivative with respect to time.
Let's break down the differentiation step by step:
1. Apply the power rule: d/dt (x^n) = n * x^(n-1).
d/dt (s^3) = 3 * s^(3-1) * (ds/dt)
2. Substitute the given rate of change of the side:
d/dt (s^3) = 3 * s^2 * (ds/dt)
Now, we'll substitute the rate of change of the side, which is given as 0.05 cm/s, into the equation:
dV/dt = 3 * s^2 * (0.05 cm/s)
So, the rate of change of the volume of the cube is given by dV/dt = 0.15s^2 cm^3/s.