Question

Volume: Given a cube with each edge expanding at a rate of 3 cm/second

How fast is the volume changing when x=1?

Answer 1

if x is the side of the cube, the volume is x^3.

so we have\[V = x^3\]
The beauty of this is that taking the derivative of both sides holds true. (related rates)
since x is unknown, we treat it as an unknown function x(t). this means we have to use chain rule.

\[\frac{ d }{ dt }V = \frac{ d }{ dt }[(x(t))^3]\]\[\frac{ dV }{ dt } = 3x^2 * \frac{ dx }{ dt }\]

you are told that dx/dt = +3 cm/s and you want to find dV/dt at x = 1. plug it in:
\[\frac{ dV }{ dt } = 3(1)^3 * 3 =\ 9 \ \frac{ cm^3 }{ s }\]

Answer 2

last line should say:
\[3(1)^2 * 3\]

and we are assuming that x = 1 CM ( they didn't give units )

Answer 3

So at the end you just plug in the 1?