An ant was randomly put somewhere on the surface of a regular tetrahedron with an edge length of 1 foot. A crumb was then randomly placed on a vertex of the tetrahedron. What is the probability that the ant can reach the crumb by walking less than foot along the surface?

Guest Jun 5, 2023

#1**0 **

The probability that the ant can reach the crumb is 0. The ant can only reach the crumb if it is placed on one of the four faces that share the vertex where the crumb is located. The probability of this happening is 1/4. However, the ant can only reach the crumb if it is within 1 foot of the vertex. The area of a face of a regular tetrahedron with an edge length of 1 foot is sqrt(2)/2. The area of a circle with a radius of 1 foot is pi. The probability that the ant is within 1 foot of the vertex is pi / (sqrt(2)/2) = 2pi. Therefore, the probability that the ant can reach the crumb is 1/4 * 2pi = pi/2.

Guest Jun 9, 2023