Two cyclists leave from an intersection at the same time. one travels due north at a speed of 18 mph, and the other travels due east at a speed of 24 mph. how long until the distance between them is 150 miles?

The ratio of speeds is 3 : 4 with a multiplier of 6 mph. They are at right angles to each other, so the speed vectors form a right triangle with side ratios 3:4:5. The separation speed of the cyclists is 5·(6 mph) = 30 mph. It will take them 5 hours to be separated by 150 miles.

time = distance/speed = 150 mi/(30 mi/h) = 5 h

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Alternate solution

The distance (d) between the cyclists can be written in terms of their respective distances as a function of time. If the first one has distance n(t) = 18t from the start, and the second has distance e(t) = 24t from the start, then the distance between them is given by the Pythagorean theorem as ...

[tex]d=\sqrt{n(t)^2+e(t)^2}=\sqrt{(18t)^2+(24t)^2}=\sqrt{900t^2}=30t\\\\150=30t \quad\text{substitute desired distance}\\\\t=\dfrac{150}{30}=\bf{5 \dots\textbf{hours}}[/tex]