Question

How fast is the edge length of a cube increasing when the volume of the cube is increasing at a rate 144cm^3/sec and the edge of length is 4cm?

Answer 1

So the formula for the volume of a cube is
\[ V = \frac{4}{3}\pi r^3\]

Can you take the differential of that?

Answer 2

wrt time

Answer 3

what is wrt?

Answer 4

with respect to

Answer 5

oh okay

Answer 6

is the volume of a cube is a^3

Answer 7

holds all the horses. I'm dumb. You're doing cubes, not spheres.

Answer 8

i was wondering about that .. lol

Answer 9

yeah, sorry about that. Autopilot for a sec. So

\[ V = a^3 \]

If you take the derivative of that with respect to time, and treat volume and edge length as functions of time (just so that they don't go to zero when you do it), what does that give you?

Answer 10

it would be dv/dt=3a?

Answer 11

i mean 3a^2

Answer 12

Super close. Remember that you need one more term multiplying what you have on the right hand side. What you got is what you would get if the whole right hand side were t^3.

Answer 13

do I multiply 3a^2 with 4? to get the dv/dt?

Answer 14

Where did the 4 come from?

It's like if you had

\[ y = x^2\]
\[ \frac{dy}{dy} = \frac{d(x^2)}{dy}\]
\[ 1 = 2x\frac{dx}{dy}\]

Answer 15

oh wait i got something... it should be like this
144=3(4)^2 x dl/dt (length)
???

Answer 16

Yah, exactly. The dl/dt is the rate that the length is changing (above I used "a"), and the V = 144cm^3/s and 'l = 4cm are the instantaneous values for dV/dt and l respectively.

Answer 17

So in the equation that you posted, what would be the part that you're looking for?

Answer 18

dl/dt!

Answer 19

Yup yup yup ^_^

Answer 20

It's applying the chain rule ^_^

Answer 21

Quick question, what is instantenous value? I heard this from my teacher, but sure what it meant

Answer 22

So when you have a rate, like the one above, the function defines the entire system for any time that you can think of. And because all of the parts are linked together, each specific value for the rate (in the example above, you're looking at exactly the point when the rate of change of the volume is 144cm^3/s) has a corresponding rate of change of the length of a leg, given the length of the leg.

Sooooo, if we held the rate of change of volume the same, but said that the length of the leg were much, much longer, the rate of change of the leg would be much smaller as well, because if it were increasing at the amount that it had been given a tiny leg, the volume would be growing REALLY fast.

Answer 23

So, an "instantaneous" is just a value at a specific point of time in the equation. In our case, we were looking at the exact "instant" the volume was increasing at 144cm^3/s, given that the length of the legs were 4cm.

A lotta text, sorry.

Answer 24

No, don't be sorry. You explain it very well, better than my teacher. I like a lot info! :) You helped me to understand the concept better! Thank you so much! :D

Answer 25

Very very welcome. Happy mathing! ^_^