Problem number three. A cylindrical can has height 15 centimeters and the perimeter of its circular base is 30 centimeters. An ant walks from point A on the base to point B on the top. Point B is directly above point A. The ant's path is either vertically upwards or horizontally along circular arcs around the can. Its path is shown with a thicker line, so black for the path on the front of the can and gray at the back. What is the length in centimeters of the ant's path? So to solve this problem we're gonna count the horizontal and the vertical parts separately so that we can really just add up the horizontal parts and the vertical parts and then add those two numbers together instead of switching between vertical, horizontal, vertical, and horizontal. We notice that the lowest horizontal path plus the highest horizontal path, so right here plus right here, they make one complete circuit. So one complete walk around the base and that's equal to the perimeter of the base of the can. The other two horizontal paths also do that. They make one complete circuit. Now the vertical paths, since B is directly above A, they're just equal to the height of the can since we're just going up once vertically one-fourth of the way four times. And so to find the total length of the path we do 2 times the perimeter of the base, so 2 times 30, plus 15, which is the height, and that gives us 75 centimeters as our answer, which is choice E.

ant path

cylindrical can

path length

perimeter

vertical component