So I saw that honorary math challenge thread and I decided to make a calculus-based one bouncing off of the other thread, because, let's face it, calculus is fun.

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1. A typical tennis ball cannister has three tennis balls. Find an equation for the area of the space in the cannister not occupied by the tennis balls in relation to the circumference of a tennis ball.

2. The most common wheel size for a standard mountain bike has an outside tire diameter of about 26.2 inches. Cross-country mountain bikes typically use the 700c standard wheel which has an outside diameter close to 28.5 inches. The mountain-bikers travel over a hill defined by the equation y = -x^2+50, where y is height in meters. If the cross-country mountain bike takes 3 minutes to travel a quarter of the way up the hill, how long will it take the standard mountain bike to get back to y=0 on the other side, assuming both bikes' wheels rotate at the same rate?

3. [Edited. Thank you mdk.] Presently, the National Hockey League (NHL) plays an 82-game schedule. Instead of winning percentages, the NHL regular season standings are based on a point system. Two points are awarded for a win, one point for losing in overtime or a shootout, and zero points for a loss in regulation. The Canucks gradually get worse as the season goes on at an exponential rate, and never go into overtime or shootouts. If they finish the season with a total of 42 points, what game number was their last win?

4. I'm thinking of joining the US Navy. Thoughts?

4. Be prepared to be on a dinghy with a bunch of seamen for a while.

Also if you're serious about joining, (like if it goes past the thinking about it step), then let me be the first to thank you for supporting our country.

Queen Raidne said 4. I'm thinking of joining the US Navy. Thoughts?

4. My brother is joining the Army... He'll be here shortly :)

4. I think you should do it, I'm an enlisted soldier in the army, so as you can see, the navy wasn't my choice.

However, disregarding that, the navy offers many jobs that can better anyone's life, though I've heard you need a high score on the ASVAB, so study and keep your confidence. From a future soldier to another possible recruit, I salute you.

And there we have it... Wise words from my bro. :)

3. Impossible scenario, since you can't get to 41 points without at least one OT loss.

.....it's the only answer I provided in the other thread and the only one I feel obligated to provide in this one :/

oh and 4, think about it. The only thing worse than not giving it a chance, would be to jump in without considering the challenges. Generally my advice is 'Do it,' but, you know. Be sure.

I looked.

I thought about it.

I remembered I haven't mathed since...like, 2000 when I got out of highschool.

I run away.

I woke up about 5 minutes ago, so forgive me if I make some stupid mistakes.

1. Not sure you need calculus for this one. By "the area of the space" I'm assuming you mean volume? And I'm assuming the canister is cylindrical.
Volume of cylinder: pir^2h, where h is the height of the cylinder.
Volume of tennis ball: Spherical, so it's 4/3pir^3. => Volume of 3 tennis balls: 4pir^3
Height of cylinder, assuming the ends of the canister are tangents to the curves of the outermost tennis balls: 3D, or 6r. Volume of cylinder is therefore: 6pir^3.
Volume of space unoccupied by balls: 6pir^3 - 4pir^3 = 2pir^3.
2pir^3 = 2pirr^2 = piDr^2 = Cr^2 (where C and D are circumference and diameter.)

2. A few questions before I start this one: firstly, by "a quarter" of the way up, do you mean in terms of displacement or distance, or in some other way? And are we considering gravitational or frictional forces, or treating this as a non-physical scenario where the example of the cyclist and the hill is for convenience's sake?

3. So, we know they won 21 games and lost 61. Other than that - and this is probably just another example of me being incredibly stupid this soon after waking up - I think we need more information. Maybe it'd be possible to set up a general equation if I thought about it, but deciphering the specifics would be a case of trial and error, I think. Please, someone correct me if I'm wrong and I'll have a proper crack at it.

4. I agree with MDK. For some people, it's a life-changing experience that does them an enormous amount of good. For others, it's the most miserable and damning experience they'll ever have. It only suits certain personality types, and you have to really be aware of what you'll sacrifice to do it. If you feel it's for you, go for it - but put some very long, hard, careful thought into it before committing.

Well then. I see how it is.