Related Rates of change (pebble/ripple problem)

[Femme_physics]

I'm not sure whether that's considered physics, algebra or calculus, but it might include all three that I just thought to post it here.

Homework Statement

http://img824.imageshack.us/img824/924/ripples.jpg

A radius of a ripple is increasing at a rate of 6 inches per second

Find how the area of the circle is changing when time = 2 sec

The Attempt at a Solution

http://img820.imageshack.us/img820/5668/pebble.jpg

 

[SteamKing]

I think the problem is asking you to calculate dA/dt when t = 2 sec. From the diagram, you also know that r = 6 inches when r = 1 s, and r = 12 inches when t = 2 s.

 

[HallsofIvy]

The problem asked for the rate of change of the area of the circle. I see nowhere in there a formula for area.

 

[I like Serena]

Hey Fp!

Your formulas are for circumferences and not for areas of circles.
Although the proper formula does have a circumference-like component in it.

I think you need to treat this as a calculus question.
Let's see if I can "step" you through this problem (if you want).

You need the formula $A = \pi R^2$ for the area of a circle.
And take the derivative as ST suggested.
That is $\frac {dA} {dt} = 2 \pi R \frac {dR} {dt}$
(As you can see this formula does contain a circumference-like component in it. )

Now you only need to fill in a value for R corresponding to t = 2 seconds, and you need to fill in the rate that the radius increases.

 

[Femme_physics]

Yes *smacks forehead*..used the wrong formula for area! Was confusing it with perimeter.

So this should make slightly more sense... although when u take the derivative of area u get perimeter! Hmmm!

http://img190.imageshack.us/img190/9265/bestshot.jpg

 

[I like Serena]

Yeah, well, taking the derivative is slightly more complex here.
But I'm not sure you've heard of the chain-rule yet...
Did you?

And are you familiar with the notation $\frac {dA} {dt}$?
This denotes the derivative of the area to the time.

The point is that you do not take the derivative to R, but you take the derivative to t.
This means that the resulting formula has to be multiplied with the derivative of R to t, that is, the rate that R increases.

 

[Femme_physics]

Damn, I'm more terrible in calculus than I thought! I just thought to practice it a bit as it would help me in more advanced dynamics!

Yeah, well, taking the derivative is slightly more complex here.
But I'm not sure you've heard of the chain-rule yet...
Did you?

I think it goes something like "the first term times the derivative of the second minus the second term times the derivative of the first".

nd are you familiar with the notation dAdt?
This denotes the derivative of the area to the time.

Oh yes, Leibovitz notation if I'm not mistaken!

This denotes the derivative of the area to the time.

Right! And we should take the derivative WITH RESPECT TO TIME because we're looking at how things change over time!

The point is that you do not take the derivative to R, but you take the derivative to t.

That means I should have an equation with R = (something)t ?

This means that the resulting formula has to be multiplied with the derivative of R to t, that is, the rate that R increases.

Hmm...I'll mull over that.

 

[I like Serena]

Damn, I'm more terrible in calculus than I thought! I just thought to practice it a bit as it would help me in more advanced dynamics!

Yes it would!

So this should make slightly more sense... although when u take the derivative of area u get perimeter! Hmmm!

Very true! Neat isn't it?

I think it goes something like "the first term times the derivative of the second minus the second term times the derivative of the first".

Uhh, no. I think that is the first part of the quotient rule.

The chain rule goes something like "the derivative of the surrounding function, multiplied by the derivative of the enclosed function".

In symbols: (u(v))' = u'(v) . v'

Oh yes, Leibovitz notation if I'm not mistaken!

Yes!
(That is, Leibnitz notation. )

Right! And we should take the derivative WITH RESPECT TO TIME because we're looking at how things change over time!

Exactly! I'm getting all excited here!

That means I should have an equation with R = (something)t ?

Yes, I think that "(something)" is given as part of your problem...

Hmm...I'll mull over that.

You do that.

 

[Mark44]

Oh yes, Leibovitz notation if I'm not mistaken!
.

(That is, Leibnitz notation. )

Um, that would be Leibniz, as in Gottfried Wilhelm (von) Leibniz. The "niz" syllable of his last name is pronounced as if it were "nitz."

 

[1MileCrash]

You are given $\frac{dr}{dt}$. You want $\frac{dA}{dt}$ so that you can solve it for your given time (radius.)

You are right in that the derivative of $\pi r^2$ is $2\pi r$, but we need to implicitly differentiate with respect to time. So, multiply this derivative by the rate of change of radius with respect to time.

$\frac{dA}{dt} = 2\pi r\frac{dr}{dt}$

If time is 2, radius is 12. The rate of area change is above, solving it with the value of the radius when t = 2 explains how area is changing when t = 2.

 

[I like Serena]

Um, that would be Leibniz, as in Gottfried Wilhelm (von) Leibniz. The "niz" syllable of his last name is pronounced as if it were "nitz."

Lol. I looked it up just before I wrote it down and got a hit on Leibnitz.
It turns out I was redirected to Leibniz without me noticing!

 

[Femme_physics]

Uhh, no. I think that is the first part of the quotient rule.

The chain rule goes something like "the derivative of the surrounding function, multiplied by the derivative of the enclosed function".

In symbols: (u(v))' = u'(v) . v'

Noted :)

implicitly differentiate

To my understand "implicitly differentiate" means using the chain rule. But why should we use the chain rule if there is one term (pi x R^2 ) to differentiate with respect to time?

Yes, I think that "(something)" is given as part of your problem...

That's the part I'm struggling with atm.

 

[chiro]

Noted :)
To my understand "implicitly differentiate" means using the chain rule. But why should we use the chain rule if there is one term (pi x R^2 ) to differentiate with respect to time?

As other people have stated above, you are finding out rates of change with respect to another quantity: namely t.

Think of a composition of functions: A is in terms of R and R is in terms of t. In terms of a "chain" this is basically A = f(R) and R = g(t). If g(t) was trivial (example g(t) = t), then dR/dt would be just 1 which means you could ignore dR/dt.

If you do multivariable calculus later (if you haven't already done it), this will help you understand what is going on. Its the same sort of idea in calculating derivatives of multivariable systems, but you deal with a matrix style approach.

 

[Femme_physics]

As other people have stated above, you are finding out rates of change with respect to another quantity: namely t.

Think of a composition of functions: A is in terms of R and R is in terms of t. In terms of a "chain" this is basically A = f(R) and R = g(t). If g(t) was trivial (example g(t) = t), then dR/dt would be just 1 which means you could ignore dR/dt.

If you do multivariable calculus later (if you haven't already done it), this will help you understand what is going on. Its the same sort of idea in calculating derivatives of multivariable systems, but you deal with a matrix style approach.

Aha, so I got TWO functions here to deal with here.

A = pi R^2
And
6R = t

Yes?

 

[chiro]

Basically dR/dt = 6 so R = 6t (initial condition is t = 0, R = 0).

This example is nice because R(t) is a simple analytic function, but you will find with later physics that your rates of change will be more complex and may not have nice analytic solutions.

But yeah you're right with A(R) = pi * R^2, and R(t) = 6t (t is in seconds, R is in inches).

 

[1MileCrash]

Because you aren't just finding the derivative, you are finding the derivative with respect to time. Area isn't an equation that involves time, this creates a new function, "with respect to time," essentially. I'll show you why it's a special case chain rule below:

The chain rule involves little work in this example, check it out:

$A = \pi r ^2$

Lets differentiate both sides, with respect to time. Set it up:

$\frac{d}{dt}[A] =\frac{d}{dt}[ \pi r ^2 ]$

What is $\frac{d}{dt} [A]$ ?

We apply the chain rule.

$\frac{dA}{dt}$ times the derivative of A, which is one.

Remember, derivative of outer function, evaluated to the inner function, times the derivative of the inner function.

So, we now have

$\frac{dA}{dt} =\frac{d}{dt}[ \pi r ^2 ]$

And yes, you can actually skip the chain rule and just move the A, it's just the "reason" why it works. You will have to actually apply the chain rule often, though, so remember it!

Now we have to differentiate the right side with respect to time. To do this, find the derivative of $[ \pi r ^2 ]$ as you've successfully done, and multiply by the derivative of radius with respect to time.$\frac{dA}{dt} =\frac{d}{dt}[ \pi r ^2 ]$
equals

$\frac{dA}{dt} = 2 \pi r \frac{dr}{dt}$

It's just about ready to solve. You know the radius you want to solve for, it's 12.

Look at
$2 \pi r \frac{dr}{dt}$

What is $\frac{dr}{dt}$?

The derivative, or rate of change of radius, with respect to time, which you are given. :)

You want the rate of change of area, with respect to time, which is sitting by itself on the left side.

You know that when time is 2, radius is 12. You know $\frac{dr}{dt}$, is just 6. You want $\frac{dA}{dt}$.

SPOILER ALERT::

$\frac{dA}{dt} = 2 \pi (12) (6)$[itex][/itex]

 

[Femme_physics]

But yeah you're right with A(R) = pi * R^2, and R(t) = 6t (t is in seconds, R is in inches).

Great! Does it mean I can just plug in the R form the second function ( R(t) = 6t) to the area function, which would turn out [ A(R) = pi x (6t)^2 ]

Would that be okay?

 

[1MileCrash]

Yes, you can do that. Albeit it's slightly unnecessary since the picture reveals that radius is 12 when time is 2. Mathematically, it's correct though.

Now derive both sides, with respect to time, solve for t = 2, and you have your answer.

 

[Femme_physics]

Because you aren't just finding the derivative, you are finding the derivative with respect to time. Area isn't an equation that involves time, this creates a new function, "with respect to time," essentially. I'll show you why it's a special case chain rule below:

The chain rule involves little work in this example, check it out:

$A = \pi r ^2$

Lets differentiate both sides, with respect to time. Set it up:

$\frac{d}{dt}[A] =\frac{d}{dt}[ \pi r ^2 ]$

What is $\frac{d}{dt} [A]$ ?

We apply the chain rule.

$\frac{dA}{dt}$ times the derivative of A, which is one.

Remember, derivative of outer function, evaluated to the inner function, times the derivative of the inner function.

So, we now have

$\frac{dA}{dt} =\frac{d}{dt}[ \pi r ^2 ]$

And yes, you can actually skip the chain rule and just move the A, it's just the "reason" why it works. You will have to actually apply the chain rule often, though, so remember it!

Now we have to differentiate the right side with respect to time. To do this, find the derivative of $[ \pi r ^2 ]$ as you've successfully done, and multiply by the derivative of radius with respect to time.$\frac{dA}{dt} =\frac{d}{dt}[ \pi r ^2 ]$
equals

$\frac{dA}{dt} = 2 \pi r \frac{dr}{dt}$

It's just about ready to solve. You know the radius you want to solve for, it's 12.

Look at
$2 \pi r \frac{dr}{dt}$

What is $\frac{dr}{dt}$?

The derivative, or rate of change of radius, with respect to time, which you are given. :)

You want the rate of change of area, with respect to time, which is sitting by itself on the left side.

You know that when time is 2, radius is 12. You know $\frac{dr}{dt}$, is just 6. You want $\frac{dA}{dt}$.

SPOILER ALERT::

$\frac{dA}{dt} = 2 \pi (12) (6)$[itex][/itex]

I am a bit confused when you do this notation

$\frac{d}{dt}[A] =\frac{d}{dt}[ \pi r ^2 ]$

A is just the function name. How can you differentiate it? I mean, I understand how to different the function itself, but why write dA/dt at the "A" side of things?

Yes, you can do that. Albeit it's slightly unnecessary since the picture reveals that radius is 12 when time is 2. Mathematically, it's correct though.

Now derive both sides, with respect to time, solve for t = 2, and you have your answer.

Oh, can I just do...

http://img204.imageshack.us/img204/2202/dadtf.jpg

Hmm but I end up with the same answer

 

[1MileCrash]

No, you cannot. It is purely coincidental. We will only arrive at the same answer when t = 2. What if they asked for area behavior when t = 3? Our formulas yield the same answer because I am multiplying 2, 6, and 12, while you are squaring 12. (12)(x) = x^2 ONLY when x is 12.

Not to get too analytical, but look at it this way, your equation shows area increasing at a rate of $\pi r^2$. That IS area. If your rate of area change equals the area formula, that would mean your area was doubling every second. A circle can't double in area every second when the radius is increasing at a constant speed! Does that make sense?

The reason why we got the same answer when t = 2, is because area in fact is doubling when t = 2, but ONLY when t = 2.

 

[Femme_physics]

BTW you asked what d/dt

Which is the derivative of (certain measurement) with respect to time

No, you cannot. It is purely coincidental. We will only arrive at the same answer when t = 2. What if they asked for area behavior when t = 3?

Then I just do

3 x 6 = 18

2 x pi x 18 = answer

 

[1MileCrash]

Then I just do

3 x 6 = 18

2 x pi x 18 = answer

That doesn't work.

 

[I like Serena]

Hey Fp!

Great! Does it mean I can just plug in the R form the second function ( R(t) = 6t) to the area function, which would turn out [ A(R) = pi x (6t)^2 ]

Would that be okay?

Yes. If you do this, you'll get the answer!

I am a bit confused when you do this notation

$\frac{d}{dt}[A] =\frac{d}{dt}[ \pi r ^2 ]$

A is just the function name. How can you differentiate it? I mean, I understand how to different the function itself, but why write dA/dt at the "A" side of things?

Differentiating is something you do with a function.
On the left hand side dA/dt shows to the reader what it is that you're differentiating.

Oh, can I just do...

Hmm but I end up with the same answer

Sorry, but you need to differentiate before plugging in the numbers.

 

[1MileCrash]

Missed this, sorry!

A is just the function name. How can you differentiate it? I mean, I understand how to different the function itself, but why write dA/dt at the "A" side of things?

A is not just the function name, A is a variable, it is just as much a differentiable function as the other side. If we differentiate one side with respect to time, we have to differentiate the other too, otherwise we just end up with a false statement.

In other words, it just means that if I ask:

f(x) = 2x + 3

You don't respond with:

f(x) = 2, because that's wrong.

you respond with
f'(x) = 2.

By putting that prime, you've shown that you're talking about the derivative of f(x), not of f(x) itself. Similarly, when I changed A to dA/dt, it means that we are no longer talking about area, I was talking about the rate that the area is changing, it's derivative. That's all I did.

Also, remember we want dA/dt when t = 2. So we should end up with dA/dt = some function with variable t or variable linked to t.

 

[Femme_physics]

Differentiating is something you do with a function.
On the left hand side dA/dt shows to the reader what it is that you're differentiating.

And what if I did dt/da? How would my differentiation be any different?

Sorry, but you need to differentiate before plugging in the numbers.

Okay, I tried, but I stopped half-way realizin g it doesn't make sense. Because when I plug 2 in the DERIVATIVE of R(t) I just get R!

http://img692.imageshack.us/img692/6064/thisiswhatigot.jpg

A is not just the function name, A is a variable, it is just as much a differentiable function as the other side. If we differentiate one side with respect to time, we have to differentiate the other too, otherwise we just end up with a false statement.

I see what you mean, we have to point out somehow that we're in fact taking the derivative and not just rewriting the function or something

 

[I like Serena]

And what if I did dt/dA? How would my differentiation be any different?

That would mean you would differentiate t with respect to A.
For that you need to have a function of A. Do you have such a function?

Okay, I tried, but I stopped half-way realizin g it doesn't make sense. Because when I plug 2 in the DERIVATIVE of R(t) I just get R!

Not quite.
Yes, you got the same number, but it has a different unit.

After 1 second you have:
R = 6 inch
And you have:
dR/dt = 6 inch/second

After 2 seconds you have:
R = 12 inch
But you still have:
dR/dt = 6 inch/second

I see what you mean, we have to point out somehow that we're in fact taking the derivative and not just rewriting the function or something

Exactly!

 

[Femme_physics]

After 1 second you have:
R = 6 inch
And you have:
dR/dt = 6 inch/second

After 2 seconds you have:
R = 12 inch
But you still have:
dR/dt = 6 inch/second

But I just did what u said, to first differentiate and later plug in. To get R = 12, I should first plug in, and then differentiate!

 

[Femme_physics]

That would mean you would differentiate t with respect to A.
For that you need to have a function of A. Do you have such a function?T (a) =...

I'm not sure what to write, so I think I don't have such function.

 

[I like Serena]

T (a) =...

I'm not sure what to write, so I think I don't have such function.

No, I guess not.

But I just did what u said, to first differentiate and later plug in. To get R = 12, I should first plug in, and then differentiate!

Keep going!

I was just trying to clarify why the 6 you got for dR/dt is not a mistake, but is to be expected.

 

[Femme_physics]

I'm not sure how to keep going, what's wrong with my last attempt?

 

[I like Serena]

Nothing is wrong with it.
It's just not finished yet.

According to the chain rule you need to calculate:
A'(R(2)) . R'(2)
As an alternative you can use what you already wrote:
A(R) = pi x (6t)^2

Take the derivative with respect to t.
And (after taking the derivative) fill in t=2.
Both ways should give you the same answer.

 

[Femme_physics]

Still same answer :(

http://img51.imageshack.us/img51/4526/dtttt.jpg

 

[I like Serena]

You forgot to take care of the round thingies.

 

[hunt_mat]

Your mistake is that you said $(6t)^{2}=36t^{2}$ and you also want:
$$\frac{dA}{dt}=\frac{dA}{dr}\frac{dr}{dt}$$

 

[1MileCrash]

Femme_Physics, what you need to understand is that we are finding the rate of change of area by observing the relationship between area and radius, because we know the rate of change of radius.

I don't see any inclusion of the rate of change of radius in your work.

I want you to solve the way you have been, when time = 10. I think you'll see a strange result that may make what I'm saying clear.