Ensign Steve

Hey, brainiacs, can you help me with this one?

In calm waters the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the area polluted is a circle and that its radius is increasing at a rate of 2 ft/sec, determine how fast the area is increasing when the radius of the circle is 40 ft.

I'm stumped.

Thanks

Bumble Bee Tuna

A= Pi*r^2
dA/dt = (dA/dr)*(dr/dt)
dA/dr = 2*Pi*r
dr/dt = 2
dA/dt = (2*Pi*40)*(2)

= 160Pi

I think.

-B

Toto

area = pi * radius squared

so if the radius is increasing at 2 ft per sec,

radius at time 1 = r
radius at time 1 plus 1 sec = r + 2

area at time 1 = pi * r squared
area at time 1 plus 1 sec = pi * (r+2) squared

So in one second the area has increased by

pi * (r squared + 4r + 4) - (pi * r squared) = pi * (4r + 4) = 164 pi

I think that's right now, as a matter of geometry not calculus

Toto

I think I would go with BBT if I were you.

Ensign Steve

Thank you sooo much! That makes perfect sense, and you got the same answer as the book! This chain rule shit is killing me. :banghead:

*pulling out hair*

Feather

Think of it as an elaborate way to multiply and divide by 1 a lot: dA/dt = dA/dt * dr/dr = dA/dr * dr/dt.

Say you have g is a function of f, which is a function of u which is a function of x. You want dg/dx. Well, dg/dx = dg/dx * df/df = dg/dx * df/df * du/du = dg/df * df/du * du/dx.

And so on. You just have to judiciously choose which differentials in the denominator you want to shuffle around and go from there.

Bumble Bee Tuna

No problem. Glad to help, Calculus sucks. I can't say it didn't take me a bit to understand WTF Feather was saying, so I will give my interpretation of the message he intended to get across-

Just treat the "dr" type things as cancellable parts of fractions (even though they technically aren't). (dA/dr)*(dr/dt), the dr's cancel and give you dA/dt.

-B

Undercurrent

Be careful with the treating dA/dt as a fraction, though, when you get to multivariate calculus. That rule pretty much goes out the window at that point.

Ensign Steve

I have no idea what that means, and hopefully I never will. I'm a business major, not a physicist!

Thanks for all the great help. My test is on Wednesday, and I my heart attacks had gone from massive to mild with 20 or so solid hours of studying.

Ensign Steve

Great news!!! I got 120% on the test. That means I got everything right, including the extra credit. It was the highest grade in the class, and half the class failed. The instructor said he has never given a 120 score to anybody before. I don't have to go back the rest of the semester or take the final, because even if I get a zero on the final (which I plan to do), I will have an A in the class. I have confirmed with my instructor via email that I can stop going, and he'll give me an A, not an incomplete. He passed my test around to the rest of the class to show them how theirs should have looked. I wasn't embarrassed at all, I was proud!

Long story short, thank you all for your help!!