Welcome to Librarium Online!
I am hellishly confused about something, and rather than bomb Phoenix, I thought I'd put it here
As you may/may not know, I have exams next week. And things aren't looking so peachy.
Whilst for Economics, English, and Biology, things aren't going so badly, Calculus is looking like it may be an uphill battle.
The question is about Rates of Change. An example question:
A spherical balloon is being filled up at a constant rate.
At one time, the volume is 300cm^3
30 seconds later, the volume is 450cm^3
Find the increase of the surface area when the balloon has a volume of 2,000cm^3.
How do I solve this? I'd like to know how, since the exam will probably feature another question that's not dissimilar.
Again, explaining Rates of Change would be helpful.
Thanks in advance,
Last edited by ArchonFarseerGuy; November 12th, 2008 at 10:37.
The rate of change is like the acceleration in your car. Your car changes position through its speed (well, velocity, innit). It changes its speed by accelerating (or braking - negative acceleration).
Back to El Balloono, you have
...A spherical balloon is being filled up at a constant rate. (important)
...At one time, the volume is 300cm^3
...30 seconds later, the volume is 450cm^3
So the rate of change for the balloon's volume is (450-300 )/30 = 5 cm^3 per second.
From here, use the formula for the balloon's volume - differentiate it to get the change in surface area. The second derivative will be the change in radius.
A useful trick is to try to picture what's going on and see if it makes sense. When the balloon is small, a 5cm^3 gain in volume is going to be relatively significant and the balloon will look like it's inflating quickly - its radius and surface area will be changing quickly. Later on, when it's at the 2000cm^3 stage, a 5cm^3 gain is minimal and the rates of change for radius and surface area will be relatively smaller.
Good luck with your exam!
Last edited by andrewthotep; November 14th, 2008 at 20:53. Reason: stoopid smileys appearing in formulae