IB myp 4-5 MATHEMATICS – Practice Questions- All Topics
Topic :Functions–Graphing trigonometric functions
Topic :Function- Weightage : 21 %
All Questions for Topic : Representation and shape of more complex functions,Transformation of quadratic functions,Rational functions,Graphing trigonometric functions,Linear programming, including inequalities,Networks-edges and arcs, nodes/ vertices, paths,Calculating network pathways,Weighted networks,Domain and range
Question
Body temperature changes during the day. The graph below shows a cosine curve modelling the body temperature for Ingrid.
$\mathrm{B}$ is the temperature in degrees Celsius $\left({ }^{\circ} \mathrm{C}\right)$
$t$ is the time in hours after midnight.
Ingrid knows it is best to sleep for 8 to 10 hours when her body temperature is $36.5^{\circ} \mathrm{C}$ or below.
Question (a)
Suggest a sleeping schedule for Ingrid.
▶️Answer/Explanation
Ans:
8 to 10 hours within the interval 6 pm to 6 am
Question (b)
Write down the time when Ingrid’s body temperature is at a maximum and a minimum.
▶️Answer/Explanation
Ans:
Maximum at 12:00 pm
Minimum at 12:00 am
Question (c)
During the day, Ingrid’s body temperature
(B) can be modelled using the equation
$$
B=-0.5 \cos \frac{\pi}{12} t+36.5
$$
where $t$ is the time in hours after $12 \mathrm{am}$. Angles are in degrees.
Write down the amplitude and period.
▶️Answer/Explanation
Ans:
In the given equation for Ingrid’s body temperature, $B = -0.5 \cos 15t + 36.5$, we can identify the amplitude and period.
The general equation for a cosine function is of the form $y = A \cos(Bx + C) + D$, where:
$\bullet$ $A$ represents the amplitude,
$\bullet$ $B$ represents the coefficient affecting the frequency or period,
$\bullet$ $C$ represents the phase shift, and
$\bullet$ $D$ represents the vertical shift.
Comparing this general equation to the given equation $B = -0.5 \cos 15t + 36.5$, we can determine the values of amplitude and period.
Amplitude: The amplitude, denoted by $A$, is the absolute value of the coefficient multiplying the cosine function. In this case, the amplitude is $| -0.5 | = 0.5$.
Period: The period, denoted by $P$, is calculated as $P = \frac{360}{B}$, where $B$ is the coefficient affecting the frequency. In this case, the coefficient is 15, so the period is $P = \frac{360}{15} = 24$.
Therefore, the amplitude of the function is 0.5, and the period is 24 hours.