IB Mathematics AHL 3.8 Trigonometric equations AI HL Paper 2- Exam Style Questions- New Syllabus

(a)
Calculate the amplitude \( a \) using the difference between high and low tide depths divided by 2:
\( \frac{18 – 4}{2} = \frac{14}{2} = 7 \)
The value of \( a \) is \( 7 \).

(b)
Calculate the vertical shift \( d \) using the average of the high and low tide depths or the high tide minus the amplitude:
\( \frac{18 + 4}{2} = \frac{22}{2} = 11 \) or \( 18 – 7 = 11 \)
The value of \( d \) is \( 11 \).

(c)
Determine the period by noting the time between high and low tide is 6 hours 15 minutes, which is \( 6 \times 60 + 15 = 375 \) minutes:
Since one full cycle includes two high tides, multiply by 2:
\( 375 \times 2 = 750 \) minutes
The period of the function is \( 750 \) minutes.

(d)
Calculate \( b \) using the period of the function:
The period \( T = \frac{360^\circ}{b} = 750 \) minutes
\( b = \frac{360}{750} = 0.48 \) degrees per minute
Alternatively, using the cosine function at the low tide point \( t = 375 \) minutes:
\( 7 \cos(b \times 375) + 11 = 4 \)
\( 7 \cos(b \times 375) = -7 \)
\( \cos(b \times 375) = -1 \)
\( b \times 375 = 180^\circ \) (since \( \cos(180^\circ) = -1 \))
\( b = \frac{180}{375} = 0.48 \) degrees per minute
Thus, the value of \( b \) is \( 0.48 \)

(e)
Set the depth function equal to the minimum allowable depth of 6 m:
\( 7 \cos(0.48t) + 11 = 6 \)
\( 7 \cos(0.48t) = -5 \)
\( \cos(0.48t) = -\frac{5}{7} \approx -0.714286 \)
\( 0.48t = \cos^{-1}(-0.714286) \)
\( t \approx \cos^{-1}(-0.714286) / 0.48 \approx 282.468 \) minutes
Convert to hours: \( 282.468 / 60 \approx 4.7078 \) hours
\( 4 \) hours \( 42.468 \) minutes \( \approx 4 \) hours \( 42 \) minutes
Add to 06:00: \( 06:00 + 4h 42m = 10:42 \)
The latest time before 12:00 that Naomi can enter is \( 10:42 \).

(f)
Find the next time the depth is 6 m after \( t = 282.468 \):
The cosine function’s next solution occurs at \( t = 750 – 282.468 \approx 467.532 \) minutes (half period from symmetry):
Duration below 6 m: \( 467.532 – 282.468 \approx 185.064 \) minutes
The duration between 06:00 and 15:00 Naomi cannot enter is \( 185 \) minutes (since 15:00 is 540 minutes, and the interval 282.468 to 467.532 is fully within this range).