An ant is at a crossroads of a square grid of highways spaced 1 meter apart.
The ant can walk 1 meter in 15 seconds along a highway, but off-road it can only travel half as fast.
Find the area of the region composed of all points the ant can reach in at most 30 seconds.
Note: an ant highway is just a line with no thickness.
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How does the above sentence relate
to to-day's date?
Finding the only zero-less answer to the alphametic
might be of help.
Graph y = x3
/3 - x2
Imagine a small circle sitting inside the local minimum. Gravity pulls in the negative y direction so it is quite stable there.
Now increase the size of the circle. At some point it will become too large to fit in this hollow and will be forced to roll away down into the third quadrant.
At what radius does this happen?
I have a square table that used to be perfectly stable, back when it had a foot at the end of each leg. Now the feet are missing from legs B, C, and D There's only one left at leg A.
The table measures 50 cm x 50 cm with legs 99.5 cm long. The foot is flat and attached by a swivel so that the end of leg A is a constant 0.5 cm from the ground. (In other words, the total length of leg A plus the foot is 100cm, 0.5 cm longer than each of the other three.)
The table can now rock back and forth with the foot of A and its opposite leg C in constant contact with the floor.
If B is also in contact with the floor, how far is D from the floor?
Note: consider the tips of the legs to be singular points at the corners of a square.