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 Mechanics - Question 15 - 2019 - Paper 2

To find the speed of the boat traveling from B to C, we need to determine the distance between points B and C and then use the formula for speed:

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

1. Calculating the Distance BC:

   • Using the previously calculated vector BC: $\mathbf{BC} = \begin{pmatrix} -130 \\ -840 \end{pmatrix}$
   • The magnitude of BC is: $|\mathbf{BC}| = \sqrt{(-130)^2 + (-840)^2} = \sqrt{16900 + 705600} = \sqrt{722500} = 850 \, \text{metres}$

2. Finding the Speed:

   • The time taken to travel from B to C is 50 seconds.
   • Hence, $\text{Speed} = \frac{850}{50} = 17 \, \text{m/s}$

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