Problem of the Week
Problem D and Solution
Too Far

Problem

Berenice is testing two robots that she built. She programs one robot to travel north at $24$ km/h and the other to travel east at $18$ km/hr. She programs the robots so that once they are $15$ km apart, they will stop moving. At exactly $1$ p.m., she starts both robots from the same location. If the robots work as intended, at what time will they stop moving?

Solution

Let $t$ be the time, in hours, that the two robots travel until they are $15$ km apart. Since one robot is traveling at $24$ km/h, it will travel $24t$ km in $t$ hours. Since the other robot is traveling at $18$ km/h, it will travel $18t$ km in $t$ hours.

Since one robot is traveling north and the other is traveling east, they are traveling at right angles to each other. We can represent the distances traveled in kilometres on a right-angled triangle as shown.

Using the Pythagorean Theorem, $$\begin{array}{rl}(24t{)}^2+(18t{)}^2& ={15}^2\\ 576t^2+324t^2& =225\\ 900t^2& =225\\ t^2& =\frac{225}{900}=\frac{1}{4}\end{array}$$ Thus, $t=\sqrt{{\displaystyle \frac{1}{4}}}={\displaystyle \frac{1}{2}}$, since $t>0$. Therefore, after half an hour, or at $1$:$30$ p.m., the two robots should be exactly $15$ km apart and should stop moving.