Cat Escape I
ِAsmaa lives in a large city, its streets in the form of a network divided into a number of squares with side length
. Asmaa had a cat named Simba. One day, Simba ran away from the house and went down to the street and started running from one of the intersections. Asmaa knew about her cat's escape after two hours. Can you help Asmaa to find her cat by calculating the area of the part that is expected to be gone by the cat? Suppose the cat is running at the yellow street with a steady speed of
, and at
on black colored streets. Assume that the cat will stop running after two hours.
The answer is 384.
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1 solution
Let's consider the two extreme cases, where the cat just walks along one street for 2 hours.
If it's the yellow street, then the cat will be 2 h ⋅ 8 k m / h = 1 6 k m away.
On the black street, the cat will travel a distance of 2 h ⋅ 6 k m / h = 1 2 k m .
For all other cases, the cat will reach the boundary of the region where it could be by walking along the yellow street (WLOG let's assume it walks east) for some time t [ h o u r s ] and then along a black street (WLOG north) for 2 − t hours.
It will get to the coordinate ( x , y ) = ( 8 t , 6 ( 2 − t ) ) . These coordinates describe a line with the equation y = − 4 3 x + 1 2 .
This means that the region the cat can reach is made up of 4 congruent right triangles (because we assumed it walks to the north east), each of which have an area of A = 2 1 ⋅ 1 6 ⋅ 1 2 = 9 6 , so the total area is 4 A = 3 8 4 .