Simple circle problem involving area and circumference

[astrololo]

Homework Statement

A stone is thrown into still water, forming ripples which travel from the center of disturbance in the form of circles. If the circumference of the circle which bounds the disturbed area is 10 ft and the circumference is increasing at the rate of 3 ft. per second, how fast is the area increasing ?

Homework Equations

C=2πr
and
A=π∗r^2

The Attempt at a Solution

I don't even know how to start this. I'm not supposed to use differentiation but only limiting process.[/B]

 

[Khashishi]

What is A as a function of C?
What is the definition of a limit?

 

[astrololo]

What is A as a function of C?
What is the definition of a limit?

Lim as h approaches 0 f(x+h)-f(x) divided by h

I don'T understand the other question.

 

[Mark44]

What is A as a function of C?
What is the definition of a limit?

Lim as h approaches 0 f(x+h)-f(x) divided by h

This is difference quotient definition of the derivative. It's not what Khashishi asked.

I don'T understand the other question.

Using ordinary algebra write A, the area of a circle, in terms of C, the circumference of the circle.

 

[Mark44]

I'm not supposed to use differentiation but only limiting process.

Why? This isn't stated in your problem description.

 

[astrololo]

Why? This isn't stated in your problem description.

Because the book that I'm using doesn't want us to use differentiation. I know that I could use these formulas, but I don't want too. (It's like we didn't learn them yet)

 

[astrololo]

This is difference quotient definition of the derivative. It's not what Khashishi asked.
Using ordinary algebra write A, the area of a circle, in terms of C, the circumference of the circle.

This : rate of change of area of circle = 1/2pi * Circumference

Is this what you want ?

 

[Mark44]

This : rate of change of area of circle = 1/2pi * Circumference

??
How did you get this?
And what we're asking in this question (restated below) has nothing to do with "rate of change" of anything.

Is this what you want ?

No.
Again, using ordinary algebra, write A, the area of a circle, in terms of C, the circumference of the circle.

 

[Ray Vickson]

Homework Statement

A stone is thrown into still water, forming ripples which travel from the center of disturbance in the form of circles. If the circumference of the circle which bounds the disturbed area is 10 ft and the circumference is increasing at the rate of 3 ft. per second, how fast is the area increasing ?

Homework Equations

C=2πr
and
A=π∗r^2

The Attempt at a Solution

I don't even know how to start this. I'm not supposed to use differentiation but only limiting process.[/B]

In one second the circumference goes from 10 to 13 ft. You can figure out what the initial and final areas are---you have all the formulas you need. However, if all you want is a rate of increase, things simplify a lot, but I will not spoil your fun by telling you how to do it.

 

[astrololo]

??
How did you get this?
And what we're asking in this question (restated below) has nothing to do with "rate of change" of anything.No.
Again, using ordinary algebra, write A, the area of a circle, in terms of C, the circumference of the circle.

Ok, I'm going to sleep for now. I'll continue tomorrow if it doesn't bother you.

 

[Mark44]

Ok, I'm going to sleep for now. I'll continue tomorrow if it doesn't bother you.

No, not a problem.

 

[astrololo]

??
How did you get this?
And what we're asking in this question (restated below) has nothing to do with "rate of change" of anything.No.
Again, using ordinary algebra, write A, the area of a circle, in terms of C, the circumference of the circle.

Ok, sorry for taking this long to continue this. Here : $$A=\frac{C^2}{4*\pi}$$

 

[Mark44]

Ok, sorry for taking this long to continue this. Here : $$A=\frac{C^2}{4*\pi}$$

Yes, that's it.

For the rest of the problem, remember that $\frac{\Delta A}{\Delta C} = \frac{A(C + \Delta C) - A(C)}{\Delta C}$, so $\Delta A = ?$

 

[astrololo]

Yes, that's it.

For the rest of the problem, remember that $\frac{\Delta A}{\Delta C} = \frac{A(C + \Delta C) - A(C)}{\Delta C}$, so $\Delta A = ?$

Ok, so I evaluated the quotient and it gave me

$$\frac{2c}{4*\pi}$$

 

[SteamKing]

Ok, so I evaluated the quotient and it gave me

$$\frac{2c}{4*\pi}$$

You can't simplify that further?

 

[astrololo]

You can't simplify that further?

Yeah sorry about that :

$$\frac{c}{2*\pi}$$

 

[SteamKing]

Yeah sorry about that :

$$\frac{c}{2*\pi}$$

Is c supposed to be the circumference of the circle?

 

[astrololo]

Is c supposed to be the circumference of the circle?

Yeah, that's what it is.

 

[Chestermiller]

Ok, sorry for taking this long to continue this. Here : $$A=\frac{C^2}{4*\pi}$$

What happens if you take this equation and differentiate both sides of the equation with respect to time. What do you get? (Of course the left hand side of the equation will represent the rate of change of area with time)

 

[astrololo]

What happens if you take this equation and differentiate both sides of the equation with respect to time. What do you get? (Of course the left hand side of the equation will represent the rate of change of area with time)

I don't understand the question. I will get the rate of change of the the area depending of the time we're talking about.

 

[Chestermiller]

I don't understand the question. I will get the rate of change of the the area depending of the time we're talking about.

I don't understand the question. I will get the rate of change of the the area depending of the time we're talking about.

Please write an equation for the time derivative of the right hand side of your equation. You do know how to take the derivative of a function, correct?

 

[astrololo]

Please write an equation for the time derivative of the right hand side of your equation. You do know how to take the derivative of a function, correct?

Yeah, but I am finding my derivatives with limits because it's what's asked in the problem. And it's derivative of what to time ? Area ?

 

[Chestermiller]

Yeah, but I am finding my derivatives with limits because it's what's asked in the problem. And it's derivative of what to time ? Area ?

Where does it say in the problem statement that you have to do it by finding the derivatives using limits? What I've been driving at is that, if $A=\frac{C^2}{4\pi}$, then $$\frac{dA}{dt}=\frac{C}{2\pi}\frac{dC}{dt}$$
Does the mathematics make any sense to you?

 

[astrololo]

Where does it say in the problem statement that you have to do it by finding the derivatives using limits? What I've been driving at is that, if $A=\frac{C^2}{4\pi}$, then $$\frac{dA}{dt}=\frac{C}{2\pi}\frac{dC}{dt}$$
Does the mathematics make any sense to you?

It's asking me specifically to use limiting process. It's a kind of introduction to calculus problem, that's why. So I can't use the technique you showed me :/

 

[Chestermiller]

OK. Let's do it using limits. Let A(t) = the area at time t, and let A(t+Δt) = area at time t + Δt. Also, let C(t) = circumference at time t, and let C(t+Δt) be the circumference at time t+Δt. Starting with your general equation for A in terms of C, write an equation for A(t+Δt) - A(t) in terms of C(t+Δt) and C(t).

 

[astrololo]

OK. Let's do it using limits. Let A(t) = the area at time t, and let A(t+Δt) = area at time t + Δt. Also, let C(t) = circumference at time t, and let C(t+Δt) be the circumference at time t+Δt. Starting with your general equation for A in terms of C, write an equation for A(t+Δt) - A(t) in terms of C(t+Δt) and C(t).

Ok, I need to go now. I'll answer you when I get time. I'm very sorry :/

 

[SteamKing]

Yeah sorry about that :

$$\frac{c}{2*\pi}$$

Then, if c is the circumference of the circle, the expression above can be simplified further still.

 

[Chestermiller]

Then, if c is the circumference of the circle, the expression above can be simplified further still.

The problem statement implies expressing dA/dt entirely in terms of the circumference and its time derivative.

Chet

 

[astrololo]

Then, if c is the circumference of the circle, the expression above can be simplified further still.

How ? I don't know what there is to do next.

 

[SteamKing]

How ? I don't know what there is to do next.

What's the formula for the circumference of a circle?

 

[astrololo]

What's the formula for the circumference of a circle?

c=2*$\pi$*r

 

[SteamKing]

c=2*$\pi$*r

Sigh... and what happens when you divide C by 2π ?

 

[astrololo]

Sigh... and what happens when you divide C by 2π ?

You obtian the radius. Btw why are you desesperate lol

 

[Chestermiller]

I was hoping you would start working this problem using the method I was leading you to in post #25. This method involves using limits (which is the main requirement for your teacher). If you do what I suggested in post #25 (which is the first step in the derivation), I can lead you through, step by step, to the final result. But I need you to start.

Chet

 

[Chestermiller]

$$A(t)=\frac{C^2(t)}{4\pi}$$
$$A(t+Δt)=\frac{C^2(t+Δt)}{4\pi}$$
$$A(t+Δt)-A(t)=\frac{C^2(t+Δt)-C^2(t)}{4\pi}$$
Do you know how to factor the numerator of the right hand side?