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

To find the speed of the boat from B to C:

1. Find the Distance BC:

   The displacement from B to C is:

   $\mathbf{BC} = \mathbf{OC} - \mathbf{OB} = \begin{pmatrix} -340 + 210 \\ -310 - 530 \end{pmatrix} = \begin{pmatrix} -130 \\ -840 \end{pmatrix}$

   Now calculate the distance using the distance formula:

   $d = \sqrt{(-130)^2 + (-840)^2} = \sqrt{16900 + 705600} = \sqrt{722500} = 850 \text{ metres}$

2. Calculate Speed:

   Speed is given by the formula:

   $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

   The time taken for the journey is 50 seconds, thus:

   $\text{Speed} = \frac{850}{50} = 17 \text{ m/s}$

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