Modelling (linear, quadratic, exponential growth/decay, cubic, sinusoidal): notes and practice questions

The population, $P$, of a certain bacterial colony, in thousands, can be modelled by the function $P(t) = 80 \times 1.15^t + 20$, where $t$ is the time in hours.

Write down the equation of the horizontal asymptote of the graph of $P(t)$.

Write down the coordinates of the point where the graph of $P(t)$ cuts the $P$-axis.

The "LearnFast" online learning platform charges a monthly subscription fee of $\$40$ and an additional $\$7$ per premium course. If a student enrolls in a minimum of $10$ premium courses in a month, a one-time discount of $\$25$ is applied to their total bill.

This can be modelled by the following function, $L$, which gives the total cost when enrolling in a minimum of $10$ premium courses at LearnFast:

$L(x) = 7x + 15, x \ge 10$

where $x$ is the number of premium courses a student enrolls in.

Find the total cost of enrolling in $20$ premium courses at LearnFast.

Find $L^{-1}(99)$.

Another online learning platform, "SkillUp", charges a flat rate of $\$8$ per premium course, with no monthly subscription or discounts. A student must enroll in a minimum of $10$ premium courses for a direct comparison with LearnFast's discounted model.

The total cost at LearnFast is cheaper than SkillUp when $x > k$.

Find the minimum integer value of $k$.

A group of engineers is designing a new observation Ferris wheel. The height, $H(t)$, in metres, of a passenger capsule above the ground is modelled by the function $H(t) = p \cos\left(\frac{\pi}{100}t\right) + q$, where $t$ is the time in seconds after the capsule begins its ascent from the highest point.

The lowest point a capsule reaches is 2 metres above the ground, and the highest point is 32 metres above the ground. The Ferris wheel completes one full rotation in 200 seconds.

Find the values of $p$ and $q$.

Using your values from part (a), the function is $H(t) = 15 \cos\left(\frac{\pi}{100}t\right) + 17$.

(i) Find $H'(t)$.

(ii) Find $H''(t)$.

The engineers are particularly interested in the moment when the capsule's vertical speed is at its maximum, for the first time after $t=0$. This occurs at time $t=k$.

Calculate the value of $k$.

Calculate the height of the capsule at this time $k$.

If the population density of a particular species of fish is measured in a large lake and the density is categorized based on the depth of the lake, D, measured in meters and the average number of fish, F, observed per year at or above a certain depth is given by the equation:

$\log_{2}F = b - \frac{D}{2}$

$a$ If at a depth of 6 meters or shallower, the average number of fish per year is 128, find the value of $b$.

$b$ According to the model, find at what depths it's expected to have more than 60 fish.

$c$ Find according to the given model the maximum height at which at least one fish is expected to be found.

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) An engineer is designing a section of a roller coaster track. To ensure a smooth ride, the track must follow a specific curve. Three sensor readings are taken at different horizontal positions along this section of the track, giving the following height coordinates: $(1, 4)$, $(2, 7)$, and $(3, 14)$.

Assuming the track follows a quadratic path of the form $y = ax^2 + bx + c$, find the equation of this quadratic curve.

(b) A fourth sensor reading is taken at a horizontal position of $x=4$, recording a height of $y=28$. Determine if this new point lies on the quadratic curve found in part (a).

(c) Due to the discrepancy, the engineer decides that a cubic function, $y = Ax^3 + Bx^2 + Cx + D$, would be a better fit for the track. Find the equation of the cubic function that passes through all four points: $(1, 4)$, $(2, 7)$, $(3, 14)$, and $(4, 28)$.

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 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) An architect is designing the entrance to a new tunnel. The shape of the entrance is modelled by a quadratic curve, with the base of the tunnel on the x-axis. The curve has end points (0, 6) and (10, 6), and its vertex is (5, 9). Distances are measured in metres.

The quadratic curve can be expressed in the form $y = ax^2 + bx + c$ for $0 \leq x \leq 10$.

(a.i) Write down the value of $c$.

(a.ii) Hence, form two equations in terms of $a$ and $b$.

(a.iii) Hence, find the equation of the quadratic curve.

(b) Calculate the area of the tunnel entrance.

A new electronics company, 'TechFlow', launched its flagship product. Let $n$ be the number of years since the product's launch. The sales team recorded the following data for the first two years.

Calculate the percentage increase in units sold from the first year to the second year.

It is assumed that the number of units sold each year will follow a geometric sequence, $u_n$.

Write down the common ratio of the sequence.

Find an expression for $u_n$.

Find the number of units TechFlow expects to sell when $n = 15$. Express your answer to the nearest integer.

In the first year, TechFlow's production facility had a capacity of $5500$ units. The company plans to increase its production capacity by $400$ units every year.

Let $v_n$ represent the production capacity of the facility in year $n$.

Write down an expression for $v_n$.

For the first $12$ years, TechFlow ensures that all units produced are sold. Each unit sold generates a profit of $30.

Calculate the total profit generated from units sold in the first $12$ years.

When $n = k$, the number of units demanded (sales) will, for the first time, exceed the production capacity.

Find $k$.

State whether, for all $n > k$, TechFlow will consistently have sales exceeding its production capacity.

Justify your answer.

A biologist is studying the growth of a bacterial colony in a petri dish. The population of the colony, $N$, can be modelled by an exponential function

$N(t) = Ae^{kt}$

where $t$ is the time in hours since the start of the experiment, and $A$ and $k$ are constants.

At the start of the experiment, the bacterial colony has a population of 250 cells. After 4 hours, the population has grown to 800 cells.

Write down the value of $A$.

Find the value of $k$.

The "SkyGazer" is a new observation wheel in a city park. The wheel has a diameter of $70$ m. To begin the ride, a passenger enters a capsule at the lowest point on the wheel, which is $3$ m above the ground. A ride consists of multiple revolutions, and the wheel makes $2$ revolutions per minute.

The height of a capsule above the ground, $h$, measured in metres, during a ride on the SkyGazer can be modelled by the function $h(t) = -a \cos (bt) + d$, where $t$ is the time, in seconds, since a passenger began their ride.

(a) Calculate the value of

(i) $a$;

(a)(ii) $b$;

(a)(iii) $d$.

(b) A ride on the SkyGazer lasts for $10$ minutes in total.

Calculate the number of revolutions of the wheel per ride.

(c) For exactly one ride on the SkyGazer, suggest

(i) an appropriate domain for $h(t)$;

(c)(ii) an appropriate range for $h(t)$.

(d) A $20$ metre-tall building stands on the horizontal ground next to the SkyGazer.

By considering the graph of $h(t)$, determine the length of time during one revolution of the wheel for which the capsule is higher than the building.

(e) There is a plan to relocate the SkyGazer onto a taller platform which will increase the maximum height of the wheel to $75$ m. This will change the value of one parameter, $a$, $b$ or $d$, found in part (a).

(i) Identify which parameter will change.

(e)(ii) Find the new value of the parameter identified in part (e)(i).

A biologist is studying the effect of a new toxin on a bacterial culture. The number of active bacterial cells, $N$ (in thousands), remaining after exposure to a toxin dose $D$ (in mg/L) follows the relationship:

$\log_{10}N = k - D$, for some constant $k \in \mathbb{R}$.

In an experiment, it was observed that when the toxin dose was 2 mg/L, there were 1000 thousand (i.e., 1,000,000) active cells remaining.

(a) Find the value of $k$.

The relationship for this bacterial culture can also be written in the form $N = \frac{b}{10^D}$.

(b) Find the value of $b$.

(c) Given that the toxin dose $D$ is between 0.5 mg/L and 4.0 mg/L (i.e., $0.5 < D < 4.0$ ), find the range for $N$.

The effectiveness score, $S$, of a new antidote is inversely proportional to the number of active cells, $N$, remaining after toxin exposure, such that $S = \frac{1}{N}$.

(d) A new antidote was tested on a culture exposed to a dose of 4.5 mg/L. Calculate the effectiveness score, $S$, for this antidote. Give your answer to 3 significant figures.

(a) An architect is designing a decorative archway for a park entrance. The cross-section of one half of the archway is modeled. The archway is symmetrical about the y-axis.

The architect models the base section of the archway as a straight line passing through the points $(0, 2)$ and $(2, 4)$, where all units are in metres.

Find the equation of the line passing through these two points.

(b) The architect initially models the curved upper section of the archway using the following measured points:

$(2, 4)$, $(4, 5)$, $(5.5, 3)$, and $(7, 0)$.

(i) Find the equation of the least squares regression quadratic curve for these four points.

(ii) By considering the gradient of this curve when $x = 2$, explain why it may not be a good model for the archway.

(c) The architect decides that a better model for the curved section would be a quadratic curve with a maximum point at $(4.5, 5.5)$ and that passes through the endpoint $(7, 0)$.

Find the equation of this new quadratic model.

(d) Believing this to be a better model for the archway, the architect wants to estimate the volume of the solid generated by rotating this half-archway about the x-axis.

(i) Write down an expression for this estimate of the volume as a sum of two integrals.

(ii) Find the value of this estimate.

A local bakery sells three types of pastries: croissants, muffins, and danishes.

• The price of a croissant is $\$4$.

• The price of a muffin is $\$3$.

• The price of a danish is $\$5$.

During a busy morning, the bakery sold a total of 460 pastries.

The total revenue from these sales was $\$1700$.

It was also noted that there were twice as many muffins sold as croissants.

Let $c$ be the number of croissants sold, $m$ be the number of muffins sold, and $d$ be the number of danishes sold.

(a) Write down three equations that express the information given above.

(b) Find the number of each type of pastry sold.

A tech company, 'CaseCrafters', designs and sells custom phone cases. Their market research suggests that the average number of cases they will sell each month is modelled by the equation

$n = 15000 - 500x$

where $n$ represents the number of cases sold and $x$ represents the selling price, in USD, of each case.

The marketing team proposes selling the cases for $22$ USD each.

Find the average number of phone cases that this model predicts CaseCrafters will sell at this price.

Calculate CaseCrafters' average monthly income, before any expenses, at this selling price.

Hence, write down the function $R(x)$ that can be used to predict CaseCrafters' average monthly income, before expenses, at any selling price, $x$.

CaseCrafters has $8000$ USD of fixed monthly operational costs. Additionally, CaseCrafters must pay their supplier $8$ USD for each phone case.

Calculate CaseCrafters' average monthly profit if they sell each case at a price of $22$ USD.

Show that the average monthly profit for any selling price, $x$, can be found using the function $P(x) = -500x^2 + 19000x - 128000$.

Find $P'(x)$.

Show that the marketing team's selling price of $22$ USD does not maximize their average monthly profit.

CaseCrafters negotiates a new deal with their supplier. Under the new deal, the supplier agrees to discount the cost of each case based on the number of cases purchased by CaseCrafters. The cost charged by the supplier for each case can be found using the function

$C_{\text{supplier}}(n) = 8 - 0.00005n$

where $n$ represents the number of cases sold by CaseCrafters.

Find the function that can be used to find CaseCrafters' average monthly profit using the new deal from the supplier.

Hence, find the selling price, per case, that CaseCrafters should choose in order to maximize their average monthly profit under the new deal.

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.

A tech company launches a new social media app. The number of active users, $U$, can be modelled by the function

$U(t) = 800 \times k^t, t\ge 0$,

where $t$ is the number of hours since the app was launched, and $k$ is a positive constant.

Write down the value of $U(0)$.

Interpret what this value means in this context.

4 hours after the app was launched, the number of active users was 4050.

Find the value of $k$.

Find the number of active users 2 hours and 15 minutes after the app was launched.

A competitor launches a similar app, whose user base, $U_2$, can be modelled by the function

$U_2(t) = 2500 \times 1.2^t, t\ge 0$,

where $t$ is the number of hours since both apps were launched.

Find the value of $t$ when the number of users for both apps is equal.

It takes $H$ hours and $m$ minutes for the number of users of the first app to reach 15000.

Find the value of $H$ and $m$, giving $m$ as an integer.

Each user of the first app requires $1.5 \times 10^{-2}$ MB of server storage. The total available server capacity is $3.0 \times 10^5$ MB.

Determine how long it would take for the app's user base to exceed the server capacity.

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.

A large water wheel is used for irrigation. The lowest point a bucket on the wheel reaches is $0.5$ m above the water surface, and its highest point is $12.5$ m above the water surface.

(a) (i) Show that the radius of the water wheel is $6$ m.

(ii) Calculate the circumference of the water wheel.

(b) When the wheel rotates $15^\circ$, find the distance that a bucket travels along the circumference.

The height in metres, above the water surface, of a particular bucket is modelled by the function:

$h(t) = a \sin (bt) + d$, for $a, b > 0$,

where $t$ is the time, measured in minutes.

The wheel takes $12$ minutes to complete $1$ revolution.

(c) (i) Find the value of $b$.

(ii) Find the value of $d$.

(iii) Hence, write down the equation of the sinusoidal model.

(d) Use the model to find the values of $t$ when the height of this bucket is $9$ m above the water surface for $0 \le t \le 12$.

The water wheel operates for $2500$ days, and each day it rotates nonstop for $10$ hours.

(e) Calculate the total number of rotations that the water wheel has made. Give your answer in the form $a \times 10^k$ where $1 \le a < 10$ and $k$ is an integer.