IB Maths AI HL 2.4 Key graph features and optimisation

[Maximum number: 5]

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 Sl 2 4 Key Graph Features 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)

Use your answers to parts (b) and (c) to find

[ 3 ]

(i)

the maximum value of R(t);

[ 2 ]

$\quad R_{\max }=\frac{1.97684 \ldots}{0.25472 \ldots}$

=7.76 \text { hours }(7.76075 \ldots)

Note: Accept 7.70 from use of 1.98.

(ii)

the minimum value of R(t).

[ 1 ]

$R_{\text {min }}=\frac{1.16474 \ldots}{0.25472 \ldots}$

=4.57 \text { hours }(4.57258 \ldots)

Note: Accept 4.55 and 4.56 from use of rounded values.

(b)

Find, in hours, the shortest time from sunrise to sunset at point A that is predicted by this model.

[ 2 ]

shortest time between sunrise and sunset
$12.4833 \ldots-3.06249 \ldots$
=9.42 hours ( $9.420843 \ldots$ )

Note: Accept 9.44 from use of 3 sf values.

IB Maths AI HL Sl 2 4 Key Graph Features Questions question 1

Question (a)

Question (i)

Question (ii)

Question (b)