Four buoys on the surface of a large, calm lake are located at A, B, C and D with position vectors given by $\vec{OA} = \begin{pmatrix} 410 \\ 710 \end{pmatrix}, \vec{OB} = \begin{pmatrix} -210 \\ 530 \end{pmatrix}, \vec{OC} = \begin{pmatrix} -340 \\ -310 \end{pmatrix} \text{ and } \vec{OD} = \begin{pmatrix} 590 \\ -40 \end{pmatrix}.$ All values are in metres - AQA - A-Level Maths Pure - Question 15 - 2019 - Paper 2

To find the speed of the boat between points B and C:

1. Distance Calculation:
   Use the distance formula between points B and C:
   $d = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2}$
   Substitute the coordinates of B and C:
   $d = \sqrt{((-340) - (-210))^2 + ((-310) - (530))^2}$
   $= \sqrt{(-130)^2 + (-840)^2}$
   $= \sqrt{16900 + 705600} = \sqrt{722500} = 850~m$

2. Speed Calculation:
   The speed is calculated as:
   $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
   Given that the time is 50 seconds, the speed is:
   $\text{Speed} = \frac{850~m}{50~s} = 17~ms^{-1}$

Hence, the speed of the boat between B and C is 17 m/s.