Developing, fitting, validating and interpreting mathematical models: notes and practice questions

The population of a certain species of fish in a lake is decreasing due to pollution, and the population at any time $t$ (in years) after the pollution started is given by the function:

$P(t) = 500(0.8)^{t},\ \ t \geq 0$

Where $P(t)$ is the population of fish in the lake at time $t$.

$a$ Find:

$i$ The initial population of fish in the lake.

$i$ The percentage decrease in the fish population each year.

$b$ Calculate the population of fish in the lake after 4 years.

A chemical compound is dissolving in a solvent. Initially, there are 225 grams of the compound. After 15 minutes, 144 grams of the compound remain undissolved.

The mass of the undissolved compound, M grams, remaining after t minutes, can be modelled by the differential equation $\frac{dM}{dt} = -k\sqrt{M}$, where k is a positive constant.

(a) Show that $M = (15 - \frac{t}{5})^2$.

(b) Calculate the time it takes for the entire compound to dissolve.

The concentration of a certain chemical, $C$, in a solution over a period of time can be modelled using the function $C(t) = -0.005t^3 + 0.1t^2 - 0.2t + 5$, where $t$ is the time in hours after the experiment begins.

Sketch the graph of $C(t) = -0.005t^3 + 0.1t^2 - 0.2t + 5$ for $0 \le t \le 20$.

Find the concentration after $2$ hours.

Find the concentration after $15$ hours.

Find the maximum concentration and the time in hours at which this occurs.

Find the minimum concentration and the time in hours at which this occurs.

Find the times in hours when the concentration is $6$ mol/L.

(a) The intensity of light, I, from a lighthouse varies inversely with the square of the distance, d, from the lighthouse, where $d > 0$.

It is known that at a distance of 2 metres from the lighthouse, the light intensity is 50 candelas per square metre (cd m$^{-2}$).

Show that $I = \frac{200}{d^2}$.

(b) Sketch the curve of I on the axes below, showing clearly the point (2, 50).

(c) A small boat needs a light intensity greater than $0.005 \text{ cd m}^{-2}$ to navigate safely.

Find the values of d where the boat cannot navigate safely.

A drone launches a package, and its trajectory is modelled by the equation $h(x) = -0.015x^2 + 0.6x + 5$, where $h(x)$ is the height of the package in metres and $x$ is the horizontal distance in metres from the launch point.

On paper, sketch the graph of the path that the package flies for $x \ge 0$. Clearly indicate the initial height, the maximum height, and the horizontal distance when it lands.

Find the height of the package when it has travelled a horizontal distance of $15$ metres.

Find the maximum height of the package.

Find the horizontal distance at which the package lands on the ground, and explain what this value represents in the context of the problem.

A manufacturing company purchases a new specialized machine. The machine's value depreciates exponentially over time. The value of the machine, $V$, in thousands of dollars, $t$ years after its purchase, is modelled by the function $V(t) = A e^{-kt}$, for $t \ge 0$.

The initial value of the machine was 150 thousand dollars. After 3 years, its value had decreased by 40%.

(a) Find the value of $k$.

(b) Calculate the value of the machine after 5 years and 6 months, giving your answer to two decimal places.

(c) The company believes that, according to this model, the machine will always have some residual value, however small.

State a mathematical reason why the company might believe this.

(d) Write down one possible limitation of the domain of the model.

(a) TechCraft Innovations starts with an annual production of $1500$ drones. Due to market demand and efficiency improvements, the production increases by $8\%$ each year on the previous year's output.

Justify that the annual drone production over the years forms a geometric sequence, and state its common ratio.

(b) Determine TechCraft Innovations' annual drone production during the third year of operation.

(c) Find the total number of drones produced by TechCraft Innovations during the first $8$ years of operation, correct to the nearest whole number.

(d) After the initial offer, TechCraft Innovations considers an alternative production plan. They want to start with the same annual production of $1500$ drones, but with a fixed annual increase of $d$ drones. The company's financial manager agrees to this new plan if the total production over the $8$ years remains the same as it was in the original geometric plan.

Determine the value of $d$, correct to two decimal places, that satisfies the financial manager's conditions.

(e) Hence, determine the number of years for which the second plan (arithmetic) gives TechCraft Innovations a higher annual production than the first plan (geometric).

An architect is designing a parabolic arch for a new building. The shape of the arch can be modelled by the function $f(x) = ax^2 + bx + c$, where $x$ and $f(x)$ are measured in metres. The highest point of the arch (the vertex) is at $(2, 4)$. One end of the arch is at the origin $(0, 0)$, and the other end is at $(p, 0)$.

(a) Find the value of $p$.

(b) Find the value of

(i) $a$.

(ii) $b$.

(iii) $c$.

(c) Write down the equation of the axis of symmetry of the arch.

A chemical engineer is studying the concentration of an intermediate product, $C$, in a reaction vessel. The change in concentration over time $t$ (in minutes) is modelled by the second order differential equation:

$\frac{d^2C}{dt^2} + 4\frac{dC}{dt} + 3C = 0$

where $t \ge 0$. It is known that when $t = 0$, the initial concentration is $C = 0$ and the rate of change of concentration is $\frac{dC}{dt} = 2$.

Show that the system of coupled first order equations:

$\frac{dC}{dt} = y$

$\frac{dy}{dt} = -3C - 4y$

can be written as the given second order differential equation.

Find the eigenvalues of the system of coupled first order equations given in part (a).

Hence find the exact solution of the second order differential equation, given the initial conditions $C(0) = 0$ and $\frac{dC}{dt}(0) = 2$.

Sketch the graph of $C$ against $t$ for $t \ge 0$, labelling the maximum point of the graph with its coordinates.

If the concentration of the intermediate product $C$ exceeds $0.2$ arbitrary units, the reaction needs to be monitored closely. Use the model to calculate the total amount of time (in minutes) during which the reaction needs to be monitored closely.

The chemical engineer decides to monitor the reaction for 15% longer than the time found from the model in part (e).

Write down one reason, with reference to the context, to support this decision.

The diagram below shows a hot air balloon hovering at point H, $520$m vertically above a landing pad.

Point A is the point on the ground, directly below the hot air balloon.

An observer starts walking at a constant speed from point C towards point A. From point C, the observer looks upward at the hot air balloon at an angle of elevation of $30\degree$. After $20$ minutes, the observer is at point B and observes the same hot air balloon at an angle of elevation of $50\degree$.

Write down the size of the angle of depression from H to C.

Find the horizontal distance from A to C.

Calculate the distance the observer walked from C to B.

Determine the observer's average speed, in metres per hour.

A landscape architect is designing a new public park. The northern boundary of the park is modelled by the function $g(x)$, and other boundaries are straight lines. The plan view of the park is shown in the following diagram, where both axes represent distance and are measured in metres.

The function $g(x)$ models the northern boundary of the park between points B and C and is given by

$g(x) = -\frac{1}{10}x^2 + 4x + 15$, for $0 \le x \le 40$.

(i) Find $g'(x)$.

(ii) Hence find the coordinates of the point on the park's northern boundary that is furthest north.

Point A has coordinates $(0, 0)$, point B has coordinates $(0, 15)$, point C has coordinates $(40, 15)$ and point D has coordinates $(40, 0)$.

(i) Write down the integral which can be used to find the area of the shaded region representing the park.

(ii) Find the area of the park.

(i) A landscaper uses the trapezoidal rule with 4 intervals to estimate the area of the park. Calculate this estimate.

(ii) Calculate the percentage error in the landscaper's estimate.

(iii) Suggest how the landscaper might be able to reduce the error whilst still using the trapezoidal rule.

A square-shaped meditation garden, PQRS, is to be built within the park. The side [PS] lies on the southern boundary (x-axis), and the side [QR] lies on the eastern boundary of the park ($x=40$). Point Q lies on the northern boundary curve $g(x)$.

(i) Find the x-coordinate of point P for the largest area of the meditation garden.

(ii) Find the largest area of the meditation garden.

Let the function $C(w)$ represent the cost in dollars per display stand, where $w$ is the width of the stand in meters.

$C(w) = \frac{500}{w^2} + 1.2$ for $5 \le w \le 15$.

Find the range of $C$.

The function $C^{-1}$ is the inverse function of $C$.

Find $C^{-1}(15)$.

In the context of the question, interpret your answer to part (b)(i).

Write down the range of $C^{-1}$.

Emily and Liam are researching the adoption of smart home devices in a specific region to create a model predicting future usage. They collect the following data:

Emily proposes the number of devices can be modelled using quadratic regression to find a function of the form $N(x) = ax^2 + bx + c$, where $x$ is the number of years after 2000.

Find the equation of Emily's model.

Emily finds the coefficient of determination for her model is $0.97977$ to five significant figures.

State whether the coefficient of determination supports Emily's proposal. Justify your answer.

Comment on the validity of Emily's model with reference to one of the parameters in the equation.

(i) Find the value of $N'(25)$ and interpret this value in context.

(ii) By considering the changes in device adoption in the table, use the value found in part (d)(i) to comment on the validity of Emily's model.

Liam proposes that the device adoption instead follows a logistic model of the form

$G(x) = \frac{1500}{1+149 \text{e}^{-0.15x}}$

where $x$ is the number of years after 2000 and $G(x)$ is the number of devices in thousands.

State a reason why it may be valid to use Liam's proposal to predict future device adoption.

(i) Find $G'(x)$.

(ii) Hence find the year, according to Liam's model, during which the greatest device adoption growth rate occurred.

The height of a diver above the water surface after jumping from a diving board is modelled by the function

$h(t) = -4.9t^2 + 5t + 10$

where $h(t)$ is the height in metres and $t$ is the time in seconds after the diver leaves the board.

(a) Write down the height of the diving board above the water surface.

(b) Find the value of $t$ when the diver enters the water. Give your answer to three significant figures.

(c) State an appropriate domain for $t$ in this model.

On a particular day, the temperature, $T$, in degrees Celsius, in the city of Solara can be modelled by the function

$T(t) = -10 \cos(b(t-4)^\circ) + 20$, where $b \in \mathbb{R}$, $0 \le t \le 24$

and $t$ represents the number of hours after midnight.

The minimum temperature recorded is $10^\circ C$ at $04:00$ ($t=4$) and the maximum temperature is $30^\circ C$ at $16:00$ ($t=16$).

Show that the value of $b$ is $15$.

Find the temperature at $07:00$ ($t=7$).

Write down

the amplitude of $T$.

the equation of the principal axis.

An outdoor event can only proceed if the temperature is between $21^\circ C$ and $27^\circ C$ (inclusive). The event organizer uses a temperature sensor that has a possible error of $\pm 0.5^\circ C$.

The organizer wants to start setting up the event as soon as possible after $08:00$ ($t=8$).

Determine the earliest time that it could be possible for the organizer to start setting up the event, ensuring that the actual temperature is certainly at or above $21^\circ C$. Give your answer to the nearest minute.

The event takes $30$ minutes for setup and $30$ minutes for pack-down. The organizer wants to maximize the actual operating time of the event, ensuring that the temperature remains within the safe range ($21^\circ C$ to $27^\circ C$) throughout its entire duration (from the start of setup to the end of pack-down).

Determine the maximum possible length of time the event can actually operate, in hours, starting from the earliest setup time found in part (d).

A new experimental drug is administered to a patient. The concentration of the drug in the patient's bloodstream, $C$, in mg/L, $t$ hours after administration, is modelled by the function $C(t) = C_0 e^{-kt}$, where $C_0$ and $k$ are positive constants.

Initially, the concentration of the drug is $250$ mg/L. After $2$ hours, the concentration drops to $150$ mg/L.

Determine the value of $k$.

Using this model, calculate the concentration of the drug in the bloodstream $5$ hours after administration.

Based on this model, will the drug ever completely leave the patient's bloodstream? Justify your answer.

State one limitation of the domain of this model in a real-world context.

The rate of change of pollution, $\frac{dP}{dt}$ (in tonnes per day), in a protected lake is modelled by $\frac{dP}{dt} = 1 - 0.1t$, where $t$ is the time in days since monitoring began, for $0 \le t \le 15$.

(i) Find the value of $\frac{dP}{dt}$ at $t = 4$ days.

(ii) Interpret the meaning of your answer to part (a) (i) in context.

Use $\frac{dP}{dt}$ to find the value of $t$ when the pollution in the lake reaches its maximum level.

Two days after monitoring began, the pollution in the lake was $5.5$ tonnes.

Find an expression for $P$ in terms of $t$, for $0 \le t \le 15$.

Hence, find the maximum pollution level in the lake.

A second source of pollution, from a nearby factory, is modelled by $Q = 3(1.15)^t$, where $Q$ is the pollution in tonnes and $t$ is the time in days, for $0 \le t \le 15$.

Write down the initial pollution from this factory.

Each day, the pollution from the factory increases by $p$ %.

Find the value of $p$.

Find the value of $t$ when the pollution from the initial lake source and the factory are the same.

A car enthusiast, Mr. Henderson, purchases two classic cars for his collection. The first is a popular sedan, initially costing $28000, which is expected to depreciate at a rate of 8$75000, and is expected to depreciate at a rate of 15% per year.

(a) Estimate the value of the popular sedan after 4 years.

(b) Find the number of years, $k$, after which both cars will have the same estimated value. Give your answer to three significant figures.

(c) Comment on the validity of your answer to part (b).

(a) A manufacturing company uses a specialized machine to produce custom parts. The number of units produced, $N$, in one day is modelled by

$N = 40 \ln(3t+1)$, for $0 \le t \le 10$

where $t$ is the time the machine operates, in hours, on that day.

Find the time, in hours, it takes for the machine to produce 80 units in one day.

(b) The daily profit, $P$, in thousands of dollars, from producing $N$ units is given by

$P = 150(1-e^{-0.008N})$, for $P \in \mathbb{R}$.

Find the profit, in thousands of dollars, the company earns for producing 80 units.

(c) Find an expression for $P$ as a function of $t$, giving your answer in the form

$P=150(1-\frac{1}{(3t+1)^n})$

where $n$ is a number to be determined.

(d) Hence or otherwise, find the profit, in thousands of dollars, the company earns by operating the machine for 4 hours.

(e) Find the greatest profit, in thousands of dollars, the company can earn in one day.

(f) The company introduces a weekly bonus profit, $B$, in thousands of dollars, for the total units produced in one week, $D_{total}$, using

$B=100(1-\frac{1}{D_{total}+1})$

The company decides to operate the machine for the same length of time, $T$ hours, each day for 5 days a week.

They want to achieve a total weekly profit of 580 thousand dollars.

Find the value of $T$.

A civil engineer is designing a parabolic arch for a pedestrian bridge. The arch starts at ground level at the origin (0,0) and reaches its maximum height of 5 meters at a horizontal distance of 10 meters from the start. The shape of the arch can be modelled by the function $f(x) = ax^2 + bx + c$, where $f(x)$ is the height of the arch above the ground in meters and $x$ is the horizontal distance in meters from the start of the arch.

(a) Find the total horizontal span of the bridge, i.e., the x-coordinate where the arch meets the ground again.

(b) Determine the values of $a$, $b$, and $c$.

(c) Write down the equation of the axis of symmetry of the parabolic arch.