IB Maths AA HL Ahl 5 14 Hl Implicit Differentiation and Related Rates Topic Practice

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.