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AHL 2.9—HL modelling functions - IB Questionbank | RevisionDojo

Practice AHL 2.9—HL modelling functions with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

A particle moves along a straight line with displacement, $s(t)$ , in meters, from a fixed point at time $t$ seconds, modeled by

s(t)=5+4 e^{-0.03 t} \sin \left(\frac{\pi}{10} t\right)-3 \cos \left(\frac{\pi}{10} t+\frac{\pi}{6}\right)

for $0 \leq t \leq 30$ . The particle is at risk of collision when $|s(t)| \geq 6$ .

Find the initial displacement of the particle.

[1]

Write down the period of the oscillatory components.

[2]

Express the oscillatory part of $s(t)$ as a single cosine function with time-dependent amplitude.

[5]

Find the first time the particle is at risk of collision.

[4]

The velocity of the particle is $v(t)=s^{\prime}(t)$ . Find the maximum acceleration of the particle in the first 30 seconds, correct to three significant figures.

[5]

Question 2

HLPaper 2

A tidal power generator produces power, $P(t)$ , in megawatts, modeled by

P(t)=10+8 \sin \left(\frac{\pi}{6} t\right)+6 e^{-0.01 t} \cos \left(\frac{\pi}{6} t-\frac{\pi}{4}\right)

where $t$ is the time in hours since midnight, for $0 \leq t \leq 48$ . The generator is considered operational when $P(t) \geq 12$ .

Find the initial power output at $t=0$ .

[1]

Determine the period of the oscillatory components.

[2]

Express the oscillatory part of $P(t)$ as a single trigonometric function with timedependent amplitude and phase shift.

[5]

Find the first time the generator is not operational ( $P(t)<12$ ) after being operational.

[4]

The revenue generated, $R(t)$ , in thousands of dollars per hour, is modeled by $R(t)= 2 P(t)+0.01 P(t)^{2}$ . Calculate the total revenue over the first 48 hours, correct to two decimal places, using numerical integration.

[5]

Question 3

HLPaper 2

A pendulum's displacement, $x(t)$ , in meters, from its equilibrium position at time $t$ seconds is modeled by

x(t)=0.5 e^{-0.1 t} \cos \left(\frac{\pi}{2} t\right)

The pendulum is released from rest at $t=0$ .

Find the initial displacement.

[1]

Determine the period of oscillation.

[2]

Show that the maximum displacement occurs at $t=0$ .

[3]

Find the time when the displacement is first at its equilibrium position.

[3]

The energy $E(t)$ of the pendulum is proportional to $x(t)^{2}$ . Given $E(0)=2$ , find the constant of proportionality and express $E(t)$ .

[3]

Determine the rate of change of $E(t)$ at $t=1$ , correct to three significant figures.

[4]

Question 4

HLPaper 2

A chemical reaction's concentration, $C(t)$ , in $\mathrm{mol} / \mathrm{L}$ , at time $t$ seconds is

C(t)=0.1+0.05 \sin \left(\frac{\pi}{4} t+\frac{\pi}{3}\right)+0.03 e^{-0.1 t} .

Find the initial concentration.

[1]

Determine the long-term concentration.

[2]

Sketch the graph of $C(t)$ for $0 \leq t \leq 8$ .

[3]

Determine the rate of change of concentration at $t=2$ , correct to three significant figures.

[3]

Question 5

HLPaper 2

A population of bacteria, $P(t)$ , in thousands, after $t$ hours is modeled by

P(t)=\frac{100}{1+99 e^{-0.5 t}}+2 \sin \left(\frac{\pi}{10} t\right)

Find the initial population.

[1]

Determine the carrying capacity of the logistic component.

[2]

Sketch the graph of $P(t)$ for $0 \leq t \leq 20$ .

[3]

Question 6

HLPaper 2

A drone's altitude, $A(t)$ , in meters, above a field at time $t$ seconds is modeled by

A(t)=50+30 \sin \left(\frac{\pi}{20} t\right)+20 e^{-0.02 t} \cos \left(\frac{\pi}{20} t+\frac{\pi}{3}\right) .

The drone's battery life depends on its altitude, with power consumption rate $B(t)= 0.1 A(t)$ watts.

Find the period of the drone's altitude oscillations.

[2]

Express the oscillatory part as a single trigonometric function with time-dependent amplitude.

[5]

Find the total energy consumed by the drone over the first 40 seconds, correct to two decimal places.

[4]

Determine the time when the drone first reaches an altitude of 60 meters.

[4]

Question 7

HLPaper 2

A wind turbine's power output, $P(t)$ , in kilowatts, varies with wind speed, which is modeled over time $t$ hours by the function

P(t)=500+300 \sin \left(\frac{\pi}{12} t+\frac{\pi}{6}\right)+200 e^{-0.05 t} \cos \left(\frac{\pi}{12} t\right)

where $t \geq 0$ . The turbine operates efficiently when $P(t) \geq 600$ . The wind speed cycles every 24 hours.

Determine the period of the oscillatory components of $P(t)$ .

[2]

Express the oscillatory part of $P(t), 300 \sin \left(\frac{\pi}{12} t+\frac{\pi}{6}\right)+200 e^{-0.05 t} \cos \left(\frac{\pi}{12} t\right)$ , in the form $A(t) \sin \left(\frac{\pi}{12} t+\phi(t)\right)$ , where $A(t)$ and $\phi(t)$ are functions of $t$ .

[5]

Sketch the graph of $P(t)$ for $0 \leq t \leq 24$ , indicating where $P(t)=600$ .

[3]

Find the time intervals in the first 24 hours when the turbine operates efficiently.

[4]

The cost of maintenance, $C(t)$ , in dollars per hour, is modeled as $C(t)=10+ 0.02 P(t)$ . Find the total maintenance cost over the first 24 hours, correct to two decimal places, using numerical integration.

[4]

Question 8

HLPaper 2

A pendulum in a clock tower swings with a motion modeled by the angular displacement, $\theta(t)$ , in radians, from its vertical position at time $t$ seconds, given by

\theta(t)=0.2 e^{-0.02 t} \sin \left(\frac{\pi}{4} t+\frac{\pi}{3}\right)+0.1 \cos \left(\frac{\pi}{4} t\right)

for $0 \leq t \leq 20$ . The pendulum is considered to be in a "safe" range when $|\theta(t)| \leq 0.15$ radians.

Find the period of the oscillatory components of $\theta(t)$ .

[2]

Express $\theta(t)$ in the form $\theta(t)=A(t) \sin \left(\frac{\pi}{4} t+\phi(t)\right)$ , where $A(t)$ and $\phi(t)$ are functions of $t$ .

[5]

Describe how to sketch the graph of $\theta(t)$ for $0 \leq t \leq 20$ , including axes labels, points where $\theta(t)= \pm 0.15$ , and the general shape.

[3]

Determine the time intervals in the first 20 seconds when the pendulum is outside the safe range $(|\theta(t)|>0.15)$ .

[4]

The energy of the pendulum is modeled by $E(t)=k \theta(t)^{2}$ , where $k$ is a constant. Given that the initial energy is $E(0)=0.5$ joules, find $k$ and the rate of change of energy at $t=5$ seconds, correct to three significant figures.

[5]

Question 9

HLPaper 2

A bridge's oscillation amplitude, $A(t)$ , in centimeters, under wind load at time $t$ seconds is modeled by

A(t)=10 e^{-0.01 t}+8 \sin \left(\frac{\pi}{30} t\right)-6 \cos \left(\frac{\pi}{30} t\right)

The bridge is at risk when $A(t)>15$ .

Find the initial amplitude.

[1]

Express the trigonometric part as a single sine function.

[4]

Find the first time the bridge is at risk.

[4]

The energy stored in the oscillation is $E(t)=k A(t)^{2}$ . Given $E(0)=200$ , find $k$ and the rate of energy change at $t=30$ , correct to three significant figures.

[5]

Question 10

HLPaper 2

The population of a rare species, $N(t)$ , in thousands, after $t$ years is modeled by

N(t)=\frac{50}{1+49 e^{-0.4 t}}+5 \cos \left(\frac{\pi}{5} t+\frac{\pi}{6}\right)

The species is at risk when $N(t)<10$ .

Find the initial population.

[1]

Determine the carrying capacity of the logistic component.

[2]

Find the time when the population first drops below 10 thousand.

[4]

The growth rate is modeled by $N^{\prime}(t)$ . Find the maximum growth rate in the first 10 years, correct to three significant figures.

[4]

Propose a modified model $M(t)=\frac{L}{1+C e^{-k t}}+5 \cos \left(\frac{\pi}{5} t+\frac{\pi}{6}\right)$ to stabilize the population above 10 thousand, and determine $L$ .

[5]

AHL 2.9—HL modelling functions Questionbank