# Partial Derivatives and The Chain Rule

## Homework Statement

The length l, width w, and height h of a box change with time. At a certain instant the dimensions are l = 7 m and w = h = 9 m, and l and w are increasing at a rate of 6 m/s while h is decreasing at a rate of 3 m/s. At that instant find the rates at which the following quantities are changing.
(a) The Volume
(b) The Surface Area
(c) The Length of the Diagonal

L2=l2+w2+h2

## The Attempt at a Solution

I have already done part a and part b but I'm having trouble with c.
I tried to work it out and got 42m/s but this is incorrect. Some help please?

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Dick
Homework Helper
42m/s is way too high. Just by gut feeling. How did you get that?

2l(dl/dt)+2w(dw/dt)+2h(dh/dt)
for A I got 675m^3/s
for B I got 312m^2/s

Dick
Homework Helper
If L^2=l^2+w^2+h^2 then 2L(dL/dt)=2l(dl/dt)+2w(dw/dt)+2h(dh/dt). You want to solve for the dL/dt part. Don't forget to divide by the L. I didn't check the other ones. Do I need to?

No the other ones are right. I have already checked them. How do I get L to know what to divide by? Do I just plug in the original l w h to find L?

Dick