Two cylinders, each with the same radius, intersect at right angles. A 3D picture of the intersection is on the right of the image. Find the volume of the intersection between the two cylinders without using calculus.
When you leave your house in the morning, you feel the sign of snow in the air. The weather report on the radio confirms your suspicions.
Sure enough, the snow begins to fall before noon-time, and falls at a constant rate. The city sends out its first snow plow at noon which begins removing snow at a constant rate in cubic feet per minute. At 1 PM the plow has gone 2 miles. At 2 PM the plow has gone 3 miles and is still not retracing its path.
At what time did it start to snow?
Consider a circle of grass with radius R containing a sheep. We’d like the sheep to eat the grass from that circle by attaching the sheep's leash on the edge of the circle. What must be the length of the leash for the sheep to eat exactly half of the grass?
A goblin is chasing us.
This time we are on a train, heading due east at 60 kilometers per hour. The goblin starts out 30 kilometers due north of us. He runs at a constant speed, but has lost the ability to plan ahead; he always heads directly towards our current position.
He catches up to us after we have traveled 100 kilometers (that is, after 100 minutes). How fast does the goblin move?
You are blindfolded. In front of you is a square turntable with four bottles at the corners. Each might be oriented "up" or "down". You make a sequence of "moves". Each move consists of five phases:
1. A referee turns the table an unknown number of quarter-turns
2. You select "adjacent" or "opposite", and then take a pair of bottles which are either "adjacent" (90 degrees apart) or "opposite" (180), accordingly
3. You observe (by touch) the current orientation of these two bottles.
4. You turn over one or the other bottle, or neither, or both.
5. The referee tells you whether all four bottles are in the same orientation; if so, you win and the game is over
After how many moves can you guarantee that you will have won? What is your strategy?