# Simple circle problem involving area and circumference

1. Nov 18, 2015

### astrololo

1. The problem statement, all variables and given/known data
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 ?

2. Relevant equations
C=2πr
and
A=π∗r^2

3. 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.

2. Nov 18, 2015

### Khashishi

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

3. Nov 18, 2015

### astrololo

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

I don'T understand the other question.

4. Nov 18, 2015

### Staff: Mentor

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.

5. Nov 18, 2015

### Staff: Mentor

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

6. Nov 18, 2015

### astrololo

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)

7. Nov 18, 2015

### astrololo

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

Is this what you want ?

8. Nov 18, 2015

### Staff: Mentor

??
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.

9. Nov 18, 2015

### Ray Vickson

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. Nov 18, 2015

### astrololo

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

11. Nov 18, 2015

### Staff: Mentor

No, not a problem.

12. Nov 20, 2015

### astrololo

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

13. Nov 20, 2015

### Staff: Mentor

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. Nov 24, 2015

### astrololo

Ok, so I evaluated the quotient and it gave me

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

15. Nov 24, 2015

### SteamKing

Staff Emeritus
You can't simplify that further?

16. Nov 24, 2015

### astrololo

Yeah sorry about that :

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

17. Nov 24, 2015

### SteamKing

Staff Emeritus
Is c supposed to be the circumference of the circle?

18. Nov 24, 2015

### astrololo

Yeah, that's what it is.

19. Nov 24, 2015

### Staff: Mentor

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. Nov 24, 2015

### astrololo

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.