Partial Derivatives and The Chain Rule

In summary, the length, width, and height of a box are changing with time, with l = 7 m, w = h = 9 m, and l and w increasing at a rate of 6 m/s while h decreases at a rate of 3 m/s. At this instant, the rates at which the volume, surface area, and length of the diagonal are changing are 675 m^3/s, 312 m^2/s, and 42 m/s, respectively. To find the rate at which the length of the diagonal is changing, we use the equation 2L(dL/dt)=2l(dl/dt)+2w(dw/dt)+2h(dh/dt), where L
  • #1
ktobrien
27
0

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

Homework Equations


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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  • #2
42m/s is way too high. Just by gut feeling. How did you get that?
 
  • #3
2l(dl/dt)+2w(dw/dt)+2h(dh/dt)
for A I got 675m^3/s
for B I got 312m^2/s
 
  • #4
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?
 
  • #5
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?
 
  • #6
ktobrien said:
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?

Absolutely.
 
  • #7
Thanks a lot. I figured it was something simple like that.
 

1. What is the purpose of using partial derivatives?

Partial derivatives allow us to analyze how a function changes with respect to one variable while holding all other variables constant. This is especially useful in multivariable calculus, as it helps us understand the behavior of a function in different directions.

2. What is the difference between a partial derivative and a regular derivative?

A partial derivative is a derivative of a function with respect to one of its variables, while holding all other variables constant. A regular derivative is a derivative with respect to a single variable. In other words, a partial derivative considers the effect of one variable on the function, while a regular derivative considers the effect of all variables on the function.

3. What is the Chain Rule and how is it used in partial derivatives?

The Chain Rule is a formula that allows us to find the derivative of a composite function. In partial derivatives, the Chain Rule is used to find the derivative of a function with respect to a specific variable while considering the effect of other variables. It is essential in solving more complex partial derivative problems.

4. Can partial derivatives be applied to any function?

Yes, partial derivatives can be applied to any function that has multiple variables. However, the function must be continuous and differentiable in order for the partial derivatives to exist.

5. How are partial derivatives useful in real-world applications?

Partial derivatives have a wide range of applications in various fields, including physics, economics, and engineering. They can be used to optimize functions in multiple variables, such as maximizing profits in economics or minimizing energy consumption in engineering. They are also used in modeling and predicting the behavior of systems in the real world.

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