Question

Please help! Will FAN AND MEDAL!

A circle is growing so that each side is increasing at the rate of 5 cm/min. How fast is the area of the circle changing at the instant the radius is 20 cm? Include units in your answer.

Answer 1

Can you please help? @across @Directrix @ganeshie8 @jim_thompson5910 @myininaya @satellite73 @zepdrix

Answer 2

A circle has no sides. Do you mean radius?

Answer 3

Yes it wants to know the area of the circle changing at the instant the radius is 20cm

Answer 4

\[A=\pi r^2\\\frac{dA}{dt}=2\pi r\frac{dr}{dt}\]
Solve for \(dA/dt\). You know what \(r\) and \(dr/dt\) are.

Answer 5

I know r is 20 but what is dr/dt?

Answer 6

It's in the first sentence.

Answer 7

Can you please show me steps? I am not exactly sure how to do it.

Answer 8

Just substitute\[r=20\\\frac{dr}{dt}=5\]

Answer 9

so \[\frac{ d(20) }{ dt }=5\]
Like this?

Answer 10

dude\[\frac{dA}{dt}=2\pi(20)(5)=200\pi\quad\frac{\text{cm.}^2}{\text{min.}}\]

Answer 11

wait why is there an A where the r is?

Answer 12

You're looking for how fast area changes relative to radius. So you need an expression for the area of the circle in terms of its radius. That's easy. That is:\[A=\pi r^2\]. But I'm interested in how the area changes with respect to time, so I differentiate with respect to time (applying chain rule and all):\[\frac{dA}{dt}=2\pi r\frac{dr}{dt}.\]The problem tells me what \(r\) and \(dr/dt\) are, so it's just now a matter of plugging in numbers and that's it.