Simple circle problem involving area and circumference

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Homework Help Overview

The problem involves a stone thrown into still water, creating ripples that form circles. The circumference of the circle is given as 10 ft, increasing at a rate of 3 ft per second, and the task is to determine how fast the area of the circle is increasing without using differentiation, but rather a limiting process.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss expressing the area as a function of circumference and question the necessity of using limits instead of differentiation. There are attempts to clarify the relationship between area and circumference using algebra.

Discussion Status

The discussion is ongoing with various interpretations being explored. Some participants are attempting to derive expressions for area in terms of circumference, while others emphasize the need to adhere to the limiting process as specified in the problem statement. There is no explicit consensus on the approach to take.

Contextual Notes

Participants note that the problem constraints include a requirement to avoid differentiation, leading to confusion about the methods available for solving the problem. The original poster expresses uncertainty about how to proceed under these constraints.

astrololo
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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]
 
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What is A as a function of C?
What is the definition of a limit?
 
Khashishi said:
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.
 
Khashishi said:
What is A as a function of C?
What is the definition of a limit?

astrololo said:
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.
astrololo said:
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.
 
astrololo said:
I'm not supposed to use differentiation but only limiting process.
Why? This isn't stated in your problem description.
 
Mark44 said:
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)
 
Mark44 said:
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 ?
 
astrololo said:
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.

astrololo said:
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.
 
astrololo said:

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.
 
  • #10
Mark44 said:
??
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.
 
  • #11
astrololo said:
Ok, I'm going to sleep for now. I'll continue tomorrow if it doesn't bother you.
No, not a problem.
 
  • #12
Mark44 said:
??
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 : $$A=\frac{C^2}{4*\pi}$$
 
  • #13
astrololo said:
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 = ?##
 
  • #14
Mark44 said:
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

$$\frac{2c}{4*\pi}$$
 
  • #15
astrololo said:
Ok, so I evaluated the quotient and it gave me

$$\frac{2c}{4*\pi}$$
You can't simplify that further?
 
  • #16
SteamKing said:
You can't simplify that further?
Yeah sorry about that :

$$\frac{c}{2*\pi}$$
 
  • #17
astrololo said:
Yeah sorry about that :

$$\frac{c}{2*\pi}$$
Is c supposed to be the circumference of the circle?
 
  • #18
SteamKing said:
Is c supposed to be the circumference of the circle?
Yeah, that's what it is.
 
  • #19
astrololo said:
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)
 
  • #20
Chestermiller said:
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.
 
  • #21
astrololo said:
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.
astrololo said:
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?
 
  • #22
Chestermiller said:
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 ?
 
  • #23
astrololo said:
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?
 
  • #24
Chestermiller said:
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 :/
 
  • #25
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).
 
  • #26
Chestermiller said:
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 :/
 
  • #27
astrololo said:
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.
 
  • #28
SteamKing said:
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
 
  • #29
SteamKing said:
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.
 
  • #30
astrololo said:
How ? I don't know what there is to do next.
What's the formula for the circumference of a circle?
 

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