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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 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 2

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 3

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.

[2]

Write down the period of the oscillatory components.

[2]

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

[3]

The velocity of the particle is $v(t)=s'(t)$ . Find the maximum acceleration of the particle in the first 30 seconds, correct to three significant figures.

[5]

Question 4

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 $t=0$ .

[2]

Determine the period of the oscillatory components.

[2]

Express the oscillatory part of $P(t)$ as a single trigonometric function with time-dependent 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 5

HLPaper 1

It is believed that two variables, $v$ and $w$ , are related by the equation $v = k w^n$ , where $k, n \in \mathbb{R}$ . Experimental values of $v$ and $w$ are obtained. A graph of $\ln v$ against $\ln w$ shows a straight line passing through $(-1.7, 4.3)$ and $(7.1, 17.5)$ .

Find the value of $k$ and of $n$ .

[4]

Question 6

HLPaper 2

A chemical reaction's concentration, $C(t)$ , in $\text{mol L}^{-1}$ , 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 7

HLPaper 2

A scientist is conducting an experiment to observe the growth of a bacteria culture. The population at any time $t$ (in hours) is given by the equation:

$P(t) = 1000 + 300 \ln(t + 1)$

Calculate the population of bacteria at $t = 3$ hours.

[2]

Find the rate of change of the population at $t = 5$ hours.

[2]

Determine the time at which the bacteria population reaches 1500.

[2]

Question 8

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)

where $0 \leq t \leq 20$ .

Find the initial population.

[1]

Determine the carrying capacity of the logistic component of the model.

[2]

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

[3]

Question 9

HLPaper 2

Consider a population of bacteria that grows exponentially. The initial population is 100 bacteria, and the population doubles every 3 hours.

Find the population of bacteria after 9 hours.

[2]

Determine the time it takes for the population to reach 1600 bacteria.

[2]

Question 10

HLPaper 2

A local bakery calculates its profit $P$ in dollars based on the number of dozens of pastries it produces, denoted as $x$ , using the equation:

$P = x\left(k - \frac{5}{200}x^2\right)$

where $k$ is a constant representing the maximum potential profit per dozen.

Find an expression for $\frac{dP}{dx}$ in terms of $k$ and $x$ .

[2]

Hence, calculate the maximum possible profit in terms of $k$ , expressing your answer in the form $pk^{3/2}$ , where $p$ is a constant.

[3]

If the bakery's profit is $360$ when it produces 10 dozen pastries, determine the value of $k$ .

[2]

A bakery's daily profit model is given by $P = x(k - \frac{5}{200}x^2)$

$x$ is the number of dozens of pastries produced
$P(x)$ is the profit in dollars

Find the optimal number of dozens of pastries the bakery should produce each morning to maximize its profit, giving your answer to the nearest whole number.

[2]

Paper

Level

Question 1

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 2

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 3

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.

[2]

Write down the period of the oscillatory components.

[2]

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

[3]

The velocity of the particle is $v(t)=s'(t)$ . Find the maximum acceleration of the particle in the first 30 seconds, correct to three significant figures.

[5]

Question 4

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 $t=0$ .

[2]

Determine the period of the oscillatory components.

[2]

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

[5]

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

[4]

[5]

Question 5

HLPaper 1

Find the value of $k$ and of $n$ .

[4]

Question 6

HLPaper 2

A chemical reaction's concentration, $C(t)$ , in $\text{mol L}^{-1}$ , 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 7

HLPaper 2

A scientist is conducting an experiment to observe the growth of a bacteria culture. The population at any time $t$ (in hours) is given by the equation:

$P(t) = 1000 + 300 \ln(t + 1)$

Calculate the population of bacteria at $t = 3$ hours.

[2]

Find the rate of change of the population at $t = 5$ hours.

[2]

Determine the time at which the bacteria population reaches 1500.

[2]

Question 8

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)

where $0 \leq t \leq 20$ .

Find the initial population.

[1]

Determine the carrying capacity of the logistic component of the model.

[2]

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

[3]

Question 9

HLPaper 2

Consider a population of bacteria that grows exponentially. The initial population is 100 bacteria, and the population doubles every 3 hours.

Find the population of bacteria after 9 hours.

[2]

Determine the time it takes for the population to reach 1600 bacteria.

[2]

Question 10

HLPaper 2

A local bakery calculates its profit $P$ in dollars based on the number of dozens of pastries it produces, denoted as $x$ , using the equation:

$P = x\left(k - \frac{5}{200}x^2\right)$

where $k$ is a constant representing the maximum potential profit per dozen.

Find an expression for $\frac{dP}{dx}$ in terms of $k$ and $x$ .

[2]

Hence, calculate the maximum possible profit in terms of $k$ , expressing your answer in the form $pk^{3/2}$ , where $p$ is a constant.

[3]

If the bakery's profit is $360$ when it produces 10 dozen pastries, determine the value of $k$ .

[2]

A bakery's daily profit model is given by $P = x(k - \frac{5}{200}x^2)$

$x$ is the number of dozens of pastries produced
$P(x)$ is the profit in dollars

Find the optimal number of dozens of pastries the bakery should produce each morning to maximize its profit, giving your answer to the nearest whole number.

[2]

Paper

Level

Question 1

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 2

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 3

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.

[2]

Write down the period of the oscillatory components.

[2]

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

[3]

The velocity of the particle is $v(t)=s'(t)$ . Find the maximum acceleration of the particle in the first 30 seconds, correct to three significant figures.

[5]

Question 4

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 $t=0$ .

[2]

Determine the period of the oscillatory components.

[2]

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

[5]

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

[4]

[5]

Question 5

HLPaper 1

Find the value of $k$ and of $n$ .

[4]

Question 6

HLPaper 2

A chemical reaction's concentration, $C(t)$ , in $\text{mol L}^{-1}$ , 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 7

HLPaper 2

A scientist is conducting an experiment to observe the growth of a bacteria culture. The population at any time $t$ (in hours) is given by the equation:

$P(t) = 1000 + 300 \ln(t + 1)$

Calculate the population of bacteria at $t = 3$ hours.

[2]

Find the rate of change of the population at $t = 5$ hours.

[2]

Determine the time at which the bacteria population reaches 1500.

[2]

Question 8

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)

where $0 \leq t \leq 20$ .

Find the initial population.

[1]

Determine the carrying capacity of the logistic component of the model.

[2]

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

[3]

Question 9

HLPaper 2

Consider a population of bacteria that grows exponentially. The initial population is 100 bacteria, and the population doubles every 3 hours.

Find the population of bacteria after 9 hours.

[2]

Determine the time it takes for the population to reach 1600 bacteria.

[2]

Question 10

HLPaper 2

A local bakery calculates its profit $P$ in dollars based on the number of dozens of pastries it produces, denoted as $x$ , using the equation:

$P = x\left(k - \frac{5}{200}x^2\right)$

where $k$ is a constant representing the maximum potential profit per dozen.

Find an expression for $\frac{dP}{dx}$ in terms of $k$ and $x$ .

[2]

Hence, calculate the maximum possible profit in terms of $k$ , expressing your answer in the form $pk^{3/2}$ , where $p$ is a constant.

[3]

If the bakery's profit is $360$ when it produces 10 dozen pastries, determine the value of $k$ .

[2]

A bakery's daily profit model is given by $P = x(k - \frac{5}{200}x^2)$

$x$ is the number of dozens of pastries produced
$P(x)$ is the profit in dollars

Find the optimal number of dozens of pastries the bakery should produce each morning to maximize its profit, giving your answer to the nearest whole number.

[2]

AHL 2.9—HL modelling functions Questionbank