Thread: help with derivatives (applied problem)

1. help with derivatives (applied problem)

Can anyone help me figure this problem out?

Any help is appreciated! Thanks

2. What have you tried so far?

3. i answered part a with V=s^3 A=s^2...i dont know what it means to differentiate dV/dt with respect to s...

4. Originally Posted by colerelm1
i answered part a with V=s^3 A=s^2...i dont know what it means to differentiate dV/dt with respect to s...
Good!

$\displaystyle \displaystyle \frac{dV}{dt}=kA=ks^2$

Do you think k will be positive or negative? (Remember that the volume is decreasing with time.)

5. Did you mean "differentiate $\displaystyle V$ with respect to $\displaystyle t$? If so, then it's called implicit differentiation:

$\displaystyle \dfrac{dV}{dt} = 3s^2 \dfrac{ds}{dt}$

6. Originally Posted by alexmahone
Good!

$\displaystyle \displaystyle \frac{dV}{dt}=kA=ks^2$

Do you think k will be positive or negative? (Remember that the volume is decreasing with time.)
ok i know its going to be negative because its decreasing but how do I do part c? I honestly have no clue where to start this one. My thinking is that dV/dt means volume(y axis) with respect to time (x axis) related to some ds/dt where s (y axis) is being differentiated with respect to time (x axis). I know how to read leibniz notation I just can't seem to grasp an understanding of it...

7. Originally Posted by colerelm1
ok i know its going to be negative because its decreasing but how do I do part c? I honestly have no clue where to start this one. My thinking is that dV/dt means volume(y axis) with respect to time (x axis) related to some ds/dt where s (y axis) is being differentiated with respect to time (x axis). I know how to read leibniz notation I just can't seem to grasp an understanding of it...
$\displaystyle \displaystyle V=s^3$

Using the Chain Rule,

$\displaystyle \displaystyle \frac{dV}{dt}=\frac{dV}{ds}\frac{ds}{dt}=3s^2\frac {ds}{dt}$

8. Originally Posted by alexmahone
$\displaystyle \displaystyle V=s^3$

Using the Chain Rule,

$\displaystyle \displaystyle \frac{dV}{dt}=\frac{dV}{ds}\frac{ds}{dt}=3s^2\frac {ds}{dt}$
ok so for part (d) i got the answer of:

$\displaystyle ks^2 = 3s^2 ds/dt$
which is the same as $\displaystyle ds/dt = ks^2 / 3s^2$

Now, for part (e) I don't get how I would use my answer from (d) to write an equation which relates s(0) to s(1) and then use it to find t_melt in terms of the quantity s_1/s_0

Can anyone help me out?