# 請問一題原文數學問題

You and a friend live 6miles apart and are talking on the phone. You

hear a crack of thunder and 8 seconds later your friend hears the

crack. Find an equation that gives the possible places where the lightning could have occurred. (Use miles as the uint of distance . The speed of sound is about 1100 ft/sec.)

Rating

This is a 3-D problem so we set locations in x,y, and z coordinates:

Without loss of generality, your position A (-3, 0,0), your friend's location B(3,0,0), and P(x,y,z) is where the thunder occur.

distance AB=1,100*8/5,280=88/52.8 miles is fixed, call it s

According to the description, we are looking for the unknown point P such that PB-PA= s. Using distance formula we arrive at the collection

of points in 3-D space:

{P(x,y,z)| sqrt[(x-3)^2+y^2+z^2]-sqrt[(x+3)^2+y^2+z^2]= s}.

Thus the desired equation is

sqrt[(x-3)^2+y^2+z^2]-sqrt[(x+3)^2+y^2+z^2]= s

If use basic algebra to simplify it we will find {P(x,y,z)} is one piece [since x less than 0] of the surface in the form x^2/a^2-y^2/b^2+z^2/c^2=1,which is classified "Hyperbola of two sheets".

A 2-D version may help us understand P(x,y) better. It will result that {P(x,y)} is one branch of the hyperbola x^2/a^2-y^2/b^2=1, where a=s/2, c=3, and b positive such that a^2+b^2=c^2. This branch (x negative) is the projection of the answer to our problem down to xy-plane.