IB Maths AI HL 1.13 Complex forms and applications

[Maximum number: 7]

A suitable site for the landing of a spacecraft on the planet Mars is identified at a point, A. The shortest time from sunrise to sunset at point A must be found.
Radians should be used throughout this question. All values given in the question should be treated as exact.
Mars completes a full orbit of the Sun in 669 Martian days, which is one Martian year.

Figure for Question IB Maths AI HL Ahl 1 13 Hl Complex Forms and Applications Questions question 1 — IB Maths AI HL

On day t, where $t \in \mathbb{Z}$ , the length of time, in hours, from the start of the Martian day until sunrise at point A can be modelled by a function, R(t), where

R(t)=a \sin (b t)+c, t \in \mathbb{R} .

The graph of R is shown for one Martian year.

(a)

Write down $z_{1}$ and $z_{2}$ in exponential form, with a constant modulus.

[ 3 ]

at least one expression in the form $r \mathrm{e}^{g(t) \mathrm{i}}$ $z_{1}=1.5 \mathrm{e}^{(0.00939 t+2.83) \mathrm{i}}, z_{2}=1.6 \mathrm{e}^{(0.00939 t) \mathrm{i}}$

(b)

Hence or otherwise find an equation for L in the form $L(t)=p \sin (q t+r)+d$ , where $p, q, r, d \in \mathbb{R}$ .

[ 4 ]

EITHER

\begin{aligned} & z_{1}-z_{2}=1.5 \mathrm{e}^{(0.00939 t+2.83) \mathrm{i}}-1.6 \mathrm{e}^{(0.00939 t) \mathrm{i}} \\ & =\mathrm{e}^{0.00939 t \mathrm{i}}\left(1.5 \mathrm{e}^{2.83 \mathrm{i}}-1.6\right) \\ & =\mathrm{e}^{0.00939 t \mathrm{i}}\left(3.06249 \ldots \mathrm{e}^{2.99086 \ldots \mathrm{i}}\right) \end{aligned}

OR
graph of L or f
p=3.06249...
$r=-0.150729 \ldots$ OR $r=2.99086 \ldots$
Note: The p and r variables (or equivalent) must be seen.

THEN
$L(t)=3.06 \sin (0.00939 t+2.99)+12.5$ $(L(t)=3.06248 \ldots \sin (0.00939 t+2.99086 \ldots)+12.4833 \ldots)$
Note: Accept equivalent forms, e.g. $L(t)=3.06 \sin (0.00939 t-0.151)+12.5$ . Follow through on their answer to part (g) replacing 12.5.

IB Maths AI HL Ahl 1 13 Hl Complex Forms and Applications Questions question 1

Question (a)

Question (b)