Simple circle problem involving area and circumference

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SUMMARY

The discussion revolves around a mathematical problem involving the area and circumference of a circle formed by ripples in water. The circumference is given as 10 ft, increasing at a rate of 3 ft/s. Participants derive the area as a function of circumference using the formula A = C²/(4π). They explore the relationship between the rate of change of area and circumference, ultimately leading to the conclusion that the rate of change of area can be expressed as dA/dt = (C/2π) * dC/dt.

PREREQUISITES
  • Understanding of basic geometry, specifically the formulas for circumference (C = 2πr) and area (A = πr²) of a circle.
  • Familiarity with limits and the concept of derivatives in calculus.
  • Ability to manipulate algebraic expressions and factor polynomials.
  • Knowledge of the relationship between rates of change in calculus.
NEXT STEPS
  • Study the derivation of the area of a circle in terms of its circumference using algebraic methods.
  • Learn about the application of limits in calculus, particularly in finding derivatives without using traditional differentiation techniques.
  • Explore the concept of related rates in calculus to understand how different quantities change with respect to time.
  • Investigate the implications of the chain rule in calculus for functions defined in terms of other functions.
USEFUL FOR

Students in introductory calculus courses, educators teaching geometry and calculus concepts, and anyone interested in understanding the mathematical relationships between area and circumference in dynamic systems.

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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