The two boats are closest together 0.36 hours after 6:00 PM, which is 21.6 minutes after 6:00 PM.
The two boats are closest together when the distance between them is minimized.
The boat traveling due south at 20 km/h is at position (0, -20t) at time t, and the boat traveling due east at 15 km/h is at position (-15 + 15t, 0) at time t.
The distance between the two boats at time t is given by the distance formula:
D(t) = sqrt((0 - (-15 + 15t))^2 + (-20t - 0)^2)
= sqrt((15 - 15t)^2 + 400t^2)
= sqrt(225 - 450t + 225t^2 + 400t^2)
= sqrt(625t^2 - 450t + 225)
To find the time t at which the distance between the two boats is minimized, we can take the derivative of D(t) with respect to t, set it equal to 0, and solve for t:
dD/dt = (1/2)(625t^2 - 450t + 225)^(-1/2)(1250t - 450) = 0
1250t - 450 = 0
t = 450/1250
t = 0.36 hours
Therefore, the two boats are closest together 0.36 hours after 6:00 PM, which is 21.6 minutes after 6:00 PM.