A spider is chasing an ant.
The spider is crawling counterclockwise at a
speed of 701 cm/min on the circumference of a circle with a diameter of 100
cm. The ant is crawling at a speed of
700 cm/min, also counterclockwise, on
a semicircle consisting of the upper
half of the spider’s circle plus a horizontal diameter.
At the start of the chase, the ant is at the left end of the diameter
and about to crawl along it, and the
spider is at the other end of the diameter and ready to start crawling along the circumference of the circle (to
which it is restricted). They commence crawling
at the same instant.
How many complete circuits of the circle must the
spider make before it catches the ant?
Idealize the problem by treating the
spider and the ant as points.
Consider the spider position as an angle from 0 to 360.
The circle has circumference 100pi cm so it would take 100pi/701 minutes to go once around. The position at time x can then be modeled by
s(x)=360*(701x/(100pi) mod 1)
The ant path has the shorter length 50pi+100.
Model its position with a negative when its on the diameter and then an angle from 0 to 180 when its on the circle. This simplifies to
a(x)=(180-360/pi)*(14x/(2+pi) mod 1)-360/pi
Each graph is a set of parallel segments that are not quite parallel to each other.
https://www.desmos.com/calculator/lxcvkiyrsi
Desmos does not like equations that set modular equations to each other. So from here I zoomed out the x-axis and scrolled along it until clicking one equation shows the point where they intersect: (450.43202, 26.1095)
https://www.desmos.com/calculator/bmekzab7ra
Meaning 450.43202 minutes. Dividing this by (701/(100pi)) gives 1005.0725 times around so the answer is 1005 complete circuits.
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Posted by Jer
on 2025-01-09 11:02:35 |
Graphical solution
