Illustration of ships related rate. Ship "A" leaves port and heads west at 17 km/hr. Ship "B" leaves port and heads south at 35 km/hr. How fast are the ships separating 5 hours after the first ship leaves port? Click the play button to animate this problem.
Define variables: [list] [*][math]A = \text{ distance of ship A from port}[/math] [*][math]B = \text{ distance of ship B from port}[/math] [*][math]D = \text{ distance between ships}[/math] [*][math]t = \text{ time}[/math] [*][math]a' = \text{ speed of ship A} = 17 km/hr [/math] [*][math]b' = \text{ speed of ship B} = 35 km/hr[/math] [*][math]d' = \text{ rate of change of distance between ships}[/math] [/list] From Pythagorean Theorem [math]D^2 = A^2 + B^2[/math] Differentiate with respect to time [math]2 D d' = 2 A a' + 2 B b'[/math] Solve for [math]d'[/math] [math]d ' = \frac{A a' + B b'}{D}[/math] Solve distances at 5 hr. [math] A = 17*5 = 85 km[/math] [math] B = 35*(5-2.5) = 87.5 km[/math] [math] D = \sqrt{A^2 + B^2 } = 121.9887 km[/math] Then substitute for [math]d'[/math] [math]d ' = \frac{(85)(17)+ (87.5)(35)}{121.9887}=36.95 km/hr [/math]