1. ## circle word problem

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.

a) Express the radius r of this circle as a function of the time t (in seconds).

b) If A is the area of this circle as a function of the radius, A(r(x)) and interpret it.

2. Originally Posted by yoman360
A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.

a) Express the radius r of this circle as a function of the time t (in seconds).
$\frac{dr}{dt}= 60 \Rightarrow r = \int \left(\frac{dr}{dt}\right)dt= r = \int 60 dt= 60t+c$

as $r = 0$ at $t = 0 \Rightarrow c=0$

therefore $r = 60t$

3. Originally Posted by yoman360
b) If A is the area of this circle as a function of the radius, A(r(x)) and interpret it.
$A= \pi r^2$ and $r = 60t$ therefore $A= \pi (60t)^2 = 3600\pi t^2$

$A = 3600\pi t^2$ is the Area of the circle at time $t$

4. Originally Posted by pickslides
$\frac{dr}{dt}= 60 \Rightarrow r = \int \left(\frac{dr}{dt}\right)dt= r = \int 60 dt= 60t+c$

as $r = 0$ at $t = 0 \Rightarrow c=0$

therefore $r = 60t$
If the speed is constant (and I assume it is because the question has been posted in the PRE-calculus subforum) then r = 60t can immediately be written (where unit of r is cm).

5. Originally Posted by pickslides
$\frac{dr}{dt}= 60 \Rightarrow r = \int \left(\frac{dr}{dt}\right)dt= r = \int 60 dt= 60t+c$

as $r = 0$ at $t = 0 \Rightarrow c=0$

therefore $r = 60t$
We don't learn integrals in pre-cal. So I can't use integrals if we never learned about them.

6. Can anyone else do this without using integrals?

7. Originally Posted by yoman360
Can anyone else do this without using integrals?
Mr F did in post 4

8. Originally Posted by e^(i*pi)
Mr F did in post 4
well how did he get r=60t?

9. Originally Posted by yoman360
well how did he get r=60t?
The radius is extending at the rate of 60cm per second.

After 1 second, the radius is 60cm
After 2 seconds, the radius is 2 x 60cm
After 3 seconds, the radius is 3 x 60cm, etc.

So if t represents the time in seconds, the radius is t x 60cm.

This is where the formula $r = 60t$ comes from.

10. Originally Posted by Prove It
The radius is extending at the rate of 60cm per second.

After 1 second, the radius is 60cm
After 2 seconds, the radius is 2 x 60cm
After 3 seconds, the radius is 3 x 60cm, etc.

So if t represents the time in seconds, the radius is t x 60cm.

This is where the formula $r = 60t$ comes from.
Thanks!

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### a stone is dropped into a lake creating circular ripples that travel outward at a speed of 60cm/s

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