A rock thrown into a pond causes a circular ripple. if the radius of the ripple is increasing by 3ft/sec how fast is the area changing when the radius is 12 feet?

Question

anonymous · Answer

Area of a circle: $$
A = \pi r^2
$$Can you differentiate with respect to time?

anonymous · Answer

Consider that area and radius are functions of time: $$
A(t) = \pi [r(t)]^2
$$

anonymous · Answer

the derivative would be 2πrr'(t) ?

anonymous · Answer

Yes.

anonymous · Answer

$$
A'(t) = 2\pi r(t) r'(t)
$$

anonymous · Answer

Can you finish it?

anonymous · Answer

is the last equation to solve it 6π(12)?

anonymous · Answer

Yeah, you can simplify it more, but that is correct.

anonymous · Answer

so the answer would be 226.08ft/sec

anonymous · Answer

Answer is $72\pi\; \text{ft}^2/\text{s}$

anonymous · Answer

wio im lost from A′(t)=2πr(t)r′(t) what numbers go in there

anonymous · Answer

$r'(t)  = 3 \;\text{ft/s}$ and $r(t) = 12\; \text{ft}$

anonymous · Answer

okay so it would be 2π(12)(3)

anonymous · Answer

Yes.

anonymous · Answer

how did u get 72 for the answer