Calculate the Final Speed: Dog's Running Equation with Breaks

To solve this problem, let's break down the information provided:

• The dog's speed for the first 15 minutes is $42$ km/h.
• The dog then rests for 2 minutes, which doesn't contribute to the average speed calculation as there's no motion.
• It continues to run for $Y$ minutes, during which it runs at an unknown speed. We need to find this speed.
• The given average speed across the entire journey is $\frac{630 + 3xy}{y + 17}$ km/h.

The goal is to determine the speed during the last $Y$ minutes, denoted as $x$ km/h. We know:
Speed in the first segment: $42$ km/h for 15 minutes, which is $\frac{15}{60}$ hours = $\frac{1}{4}$ hours. Thus, the distance covered is:

$\text{Distance}_1 = 42 \times \frac{1}{4} = 10.5$ km

The dog runs for a total of $15 + 2 + Y = Y + 17$ minutes. In hours, this time is $\frac{Y + 17}{60}$.

The average speed formula gives us:

$\frac{\text{Total Distance}}{\text{Total Time}} = \frac{630 + 3xy}{y + 17}$

However, this expression for average speed is already given, so we equate it with the steps to form an equation:

Average speed from the total journey equation:

$\frac{10.5 + x \times \frac{y}{60}}{\frac{y + 17}{60}} = \frac{630 + 3xy}{y + 17}$

The denominators cancel out $y + 17$ implying the speed $x$ to be $3x$ given based choice matching.

This implies that the consistent representation shows the dog's speed in the last $Y$ minutes is $\textbf{equal to}$ $\frac{3xy}{y}$ which reduces to $\mathbf{3x}$ km/h under the given parameters.

Therefore, the speed during the last $Y$ minutes is $3x$ km/h.

The correct answer, corresponding to the choices given, is $\mathbf{3x}$ km/h.