Answer
The boats are closest together at approximately $~~2:22~pm$
Work Step by Step
Let $t$ be the time after $2:00~pm$
Let $x$ be the distance from the dock to the boat traveling east.
$x = 15-15t$
Let $y$ be the distance from the dock to the boat traveling south.
$y = 20t$
We can find an expression for the distance between the two boats:
$D = \sqrt{x^2+y^2}$
$D = \sqrt{(15-15t)^2+(20t)^2}$
$D = \sqrt{225t^2-450t+225+400t^2}$
$D = \sqrt{625t^2-450t+225}$
We can find $\frac{dD}{dt}$:
$\frac{dD}{dt} = \frac{1250t-450}{2\sqrt{625t^2-450t+225}} = 0$
$1250t-450 = 0$
$t = \frac{450}{1250}$
$t = 0.36$
$t \approx 22~minutes$
The boats are closest together at approximately $~~2:22~pm$
