IB Maths AA HL 5.14 Implicit differentiation and related rates

[Maximum number: 6]

Two boats A and B travel due north.
Initially, boat B is positioned 50 metres due east of boat A .
The distances travelled by boat A and boat B , after t seconds, are x metres and y metres respectively. The angle $\theta$ is the radian measure of the bearing of boat B from boat A . This information is shown on the following diagram.

Figure for Question IB Maths AA HL Ahl 5 14 Hl Implicit Differentiation and Related Rates Questions question 1 — IB Maths AA HL

Find the speed of boat A at time T.

attempt to differentiate with respect to t
$\frac{\mathrm{d} y}{\mathrm{~d} t}=\frac{\mathrm{d} x}{\mathrm{~d} t}-50(\operatorname{cosec} \theta)^{2} \frac{\mathrm{~d} \theta}{\mathrm{~d} t}$
attempt to set speed of B equal to double the speed of A
$2 \frac{\mathrm{~d} x}{\mathrm{~d} t}=\frac{\mathrm{d} x}{\mathrm{~d} t}-50(\operatorname{cosec} \theta)^{2} \frac{\mathrm{~d} \theta}{\mathrm{~d} t}$ $\frac{\mathrm{d} x}{\mathrm{~d} t}=-50(\operatorname{cosec} \theta)^{2} \frac{\mathrm{~d} \theta}{\mathrm{~d} t}$ $\theta=\arctan 5(=1.373 \ldots=78.69 \ldots)$ OR $\operatorname{cosec}^{2} \theta=1+\cot ^{2} \theta=1+\left(\frac{1}{5}\right)^{2}=\frac{26}{25}$

Note: This A1 can be awarded independently of previous marks.
$\frac{\mathrm{d} x}{\mathrm{~d} t}=-50\left(\frac{26}{25}\right) \times-0.1$
So the speed of boat A is $5.2\left(\mathrm{~ms}^{-1}\right)$

Note: Accept 5.20 from the use of inexact values.

IB Maths AA HL Ahl 5 14 Hl Implicit Differentiation and Related Rates Questions question 1