Question

A Tibetan monk leaves the monastery at 7:00 AM and takes his usual path to the top of the mountain, arriving at 7:00 PM. The following morning, he starts at 7:00 AM at the top and takes the same path back, arriving at the monastery at 7:00 PM. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.

Short answer

It is proved that there is a point on the path that monks will cross at exactly the same time of day.

Step-by-step solution

Intermediate value theorem

Consider thefunction\(f\)iscontinuous on the closed interval\(\left( {a,b} \right)\)and let\(N\)be any number between\(f\left( a \right)\)and\(f\left( b \right)\)with\(f\left( a \right) \ne f\left( b \right)\). Then there exists anumber\(c\)in \(\left( {a,b} \right)\) such that \(f\left( c \right) = N\).

Prove that there is a point on the path that the monk will cross at the same time of day

Let\(u\left( t \right)\)be the monk’s distance from the monastery as a function of time\(t\)(in hours) on the first day.

Let\(d\left( t \right)\)be his distance from the monastery as a function of time\(t\)(in hours) on the second day.

Consider\(D\)be the distance from the monastery to the top of the mountain.

It is observed from the given information that\(u\left( 0 \right) = 0,u\left( {12} \right) = D,d\left( 0 \right) = D,\)and\(d\left( {12} \right) = 0\).

Let \(u - d\) be the function, which is continuous. It is determined that \(\left( {u - d} \right)\left( 0 \right) = - D\)and \(\left( {u - d} \right)\left( {12} \right) = - D\).

Therefore, according to the Intermediate Value Theorem, there is some time \({t_0}\) between \(0\) and 12 such that \(\left( {u - d} \right)\left( {{t_0}} \right) = 0 \Leftrightarrow u\left( {{t_0}} \right) = d\left( {{t_0}} \right)\).

Hence, the monk will cross at the same place on both daysafter 7:00 AM at a time \({t_0}\).

Thus, it is proved that there is a point on the path that monks will cross at exactly the same time of day.