rate of change help

• Aug 7th 2008, 10:01 AM
TheJacksonater
rate of change help
A rectangular prisms length is is increasing by 12 meters per min, its height increasing by 2 meters per min and its width increasing by 4 meters per minute. How fast is the volume changing when the dimensions are:
length = 200 m
width = 50 m
height = 30 m

Cant seem to figure this one out. Thanks for any help
-Jackson
• Aug 7th 2008, 10:22 AM
Soroban
Hello, TheJacksonater!

Quote:

A rectangular prism`s length is increasing by 12 m/min,
its height increasing by 2 m/min and its width increasing by 4 m/min.
How fast is the volume changing when the dimensions are:
length = 200 m, width = 50 m, height = 30 m.

We are given: . $\begin{Bmatrix}
\dfrac{dL}{dt}&=&12 \\ \\[-3mm] \dfrac{dW}{dt} &=& 4 \\ \\[-3mm]
\dfrac{dH}{dt} &=& 2 \end{Bmatrix} \quad\text{ and }\quad
\begin{Bmatrix}L &=&200 \\ W &=&50 \\ H&=&30 \end{Bmatrix}$

We know that: . $V \;=\:L\cdot W\cdot H$

Differentiate with respect to time: . $\frac{dV}{dt} \;=\;L\!\cdot\!W\!\cdot\frac{dH}{dt} + L\!\cdot\!H\!\cdot\!\frac{dW}{dt} + W\!\cdot\!H\!\cdot\!\frac{dL}{dt}$

Therefore: . $\frac{dV}{dt} \;=\;(200)(50)(2) + (200)(30)(4) + (50)(30)(12) \;=\;\boxed{62,000\text{ m}^3\text{/min}}$