Question

Hello, can somebody help me? I currently need to solve a related rate problem and I want to know the general way to do it.

How do I solve for the rate of volume increasing/decreasing when the height increases at a certain rate and the length decreases at a certain rate?

Answer 1

OOOOhhhh lemmer seee

Answer 2

look and see if this kinda helps
dnichols30582.edublogs.org/files/2014/06/3.11-Solutions-2i0f245.pdf

Answer 3

www2.bc.cc.ca.us/resperic/math6a/lectures/ch4/4/HW4.4/HW4.4sols.htm

Answer 4

you might want to be specific. I can try to answer it generally but that might confuse you more.

Answer 5

oh wait you asked for generalization sec

Answer 6

How do I solve for the rate of volume increasing/decreasing when the height increases at a certain rate and the length decreases at a certain rate?

you want the rate of volume change. Or, dV/dt
V is a function of multiple things... V(l,h)

Answer 7

I'm not watching but maybe I'll comment in an hour from now when I'm done browsing dank memes

Answer 8

Bless @Kainui

Answer 9

In my case, the height is 4 and increases by 2 cm/min and the height is 15 cm and decreases at 3 cm/min. I just wanna learn the general way to do it so get a better conceptual understanding, you know?

Answer 10

what comes next might be considered multivariate calculus (I believe it's the multivariate chain rule) but let's see
generally speaking l is a function of h, or h is a function of l (say in the case of a cone, there is a Pythagorean relationship etc)
so we can rewrite as V(l,h(l))....
we want
\(\Huge \frac{dV}{dt}=\frac{\partial V}{\partial l}*\frac{dl}{dt}+\frac{\partial V}{\partial h}*\frac{dh}{dt}\)
or
\(\Huge \frac{dV}{dt}=\frac{\partial V}{\partial l}*\frac{dl}{dt}+\frac{\partial V}{\partial h}*\frac{dh}{dl}*\frac{dl}{dt}\)

Answer 11

note the multivariate chain rule states, for two variables (extendable up to n), that
if f is a function of x and y, and x and y are functions of s, then
\(\Huge \frac{df}{ds}=\frac{\partial f}{\partial x}*\frac{dx}{dt}+\frac{\partial f}{\partial y}*\frac{dy}{dt}\)
partial symbols which look like backward sixes just mean that you derive with respect to that variable in particular whilst holding other variables to be constant, for instance;
\(\Huge \frac{\partial}{\partial x}(5x^3y^2za)=15x^2y^2za\)

Answer 12

Correction: those t's should be s's sorry

Answer 13

Let me know if you need clarifications.

Answer 14

I'm trying to apply this to the math question I'm currently on. Thank you.

Answer 15

Note that if h is not a function of l than the first form
\(\Huge \frac{dV}{dt}=\frac{\partial V}{\partial l}*\frac{dl}{dt}+\frac{\partial V}{\partial h}*\frac{dh}{dt}\)
is applicatble and not the second as dh/dl is non sequitur