Related Rates of Volume Change for Expanding Cube Edges

[Michele Nunes]

Homework Statement

All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters?

Homework Equations

The Attempt at a Solution

I used the equation for the volume of a cube: V = s³ but I'm not sure if side and edge can be considered the same thing. Anyways, I implicitly differentiated it with respect to time t and got: dV/dt = 3s²(ds/dt) and since they give ds/dt = 3 cm/sec and values for s I just plugged all that in and for (a) I got 9 cm³/sec and for (b) I got 900 cm³/sec but I'm not sure if I did it correctly though.

 

[Mark44]

Homework Statement

All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters?

Homework Equations

The Attempt at a Solution

I used the equation for the volume of a cube: V = s³ but I'm not sure if side and edge can be considered the same thing. Anyways, I implicitly differentiated it with respect to time t and got: dV/dt = 3s²(ds/dt) and since they give ds/dt = 3 cm/sec and values for s I just plugged all that in and for (a) I got 9 cm³/sec and for (b) I got 900 cm³/sec but I'm not sure if I did it correctly though.

Both answers are correct.

To answer your other question, a side and an edge are the same thing here.

 

[Michele Nunes]

Both answers are correct.

To answer your other question, a side and an edge are the same thing here.

Thank you for double checking my work

 

[Mark44]

Thank you for double checking my work

You're welcome!

 

[fresh_42]

As you mentioned is $s = s(t) = 3t + s_0$ where $s_0$ is the starting length of the cube. Isn't the volume's vvelocity then accelerating quadratic in time?
And if so when will be measured? At $3t + s_0 = 1$ and $3t + s_0 = 100$ or is $s_0 = 0$, $s_0 = 100$ resp.?

 

[Mark44]

As you mentioned is $s = s(t) = 3t + s_0$ where $s_0$ is the starting length of the cube. Isn't the volume's vvelocity then accelerating quadratic in time?

No, you are misinterpreting the problem. The cube isn't moving through space. It is expanding. This is a typical problem in calculus textbooks in the section on Related Rates.

And if so when will be measured? At $3t + s_0 = 1$ and $3t + s_0 = 100$ or is $s_0 = 0$, $s_0 = 100$ resp.?

 

[fresh_42]

@Mark44 I got that. $s(t)$ has been noted the length of the cube's edges in the OP.

My misunderstanding was that I first thought the given lengths were those of the original cube when expansion started.
In that case it would have been just a formula of time and time of measurement needed to be specified.

But meanwhile I understood it: $s(t) = 3t + s_0 = 1$ or $100$ defines the measurement via the actual edges then.
I have to admit that I sometimes tend to make things more complicated than they are.
Thank you for replying.