Optimisation (DP IB Analysis & Approaches (AA): HL): Exam Questions

A company makes food tins in the shape of closed cylinders. Each tin must have a volume of $150\pi$ cm³. To reduce material costs, the company would like to minimise the surface area of the tins.

Let the radius of a tin be $r$ cm and its height be $h$ cm.

By first expressing $h$ in terms of $r$, show that the surface area, $S$ cm², of a tin is given by $S=2\pi r^2+\frac{300\pi }{r}$.

Use calculus to find the minimum surface area of a tin. Give your answer correct to two decimal places.

A right-angled triangle with height $h$ cm, base $r$ cm and hypotenuse 15 cm is rotated about its vertical side to form a cone, as shown in the following diagram.

Find an expression for $r$ in terms of $h$.

Show that the volume of the cone, $V$ cm³, is given by

$V=\frac{\pi }{3}\left(225h-h^3\right)$

Find the value of $h$ that gives the maximum volume of the cone.

A piece of wire 1 m long is cut into two pieces. The first piece is bent into the shape of a square. The second piece is bent into the shape of a rectangle whose length, $l$ cm, is twice its width, $w$ cm. The perimeter of the square is $x$ cm.

Find an expression, in terms of $x$, for the area of the square.

Show that $w=\frac{100-x}{6}$.

Find an expression, in terms of $x$, for the sum, $S$ cm², of the areas of the two shapes.

Find the value of $x$ for which $S$ is a minimum.

Liam, a keen runner and swimmer, enters a race. Competitors run from point $\mathrm{A}$ along a straight beach to a point $\mathrm{B}$, then dive into the water and swim directly to point $\mathrm{C}$. Liam runs at 8 m s^-1 along the beach and swims at 2 m s^-1 in the water.

Let $\mathrm{AB}=x$ metres, the distance that Liam runs along the beach before he swims along $\left[\mathrm{BC}\right]$.

Find an expression for the time, in seconds, that Liam takes to run from $\mathrm{A}$ to $\mathrm{B}$.

Show that $\mathrm{BC}=\sqrt{10000+{\left(500-x\right)}^2}$.

Find an expression for the total time, $T$ seconds, that Liam takes to complete the race.

Find the value of $x$ that minimises $T$.

A small cylindrical drum is closed at the top but open at the bottom. It has a radius of $r$ cm and a height of $h$ cm, and its volume is 1000 cm³.

The material for the top of the drum costs 25 cents per cm² and the material for the curved surface costs 20 cents per cm².

Find an expression for $h$ in terms of $r$.

Show that the total cost, $C$ cents, of the material to make the drum is given by $C=25\pi r^2+\frac{40000}{r}$.

Find $\frac{\mathrm{d}C}{\mathrm{d}r}$.

The function $C\left(r\right)$ has a local minimum at the point $(p,q)$.

Find the value of $p$.

State, to the nearest dollar, the minimum cost of the material to make the drum.

Find $\frac{{\mathrm{d}}^{2}C}{\mathrm{d}{r}^2}$ and hence describe the concavity of the graph of $C\left(r\right)$ at $r=p$.

A company produces and sells cricket bats. The company's daily cost, $C$, in hundreds of Australian dollars (AUD), of producing $x$ hundred cricket bats in a day can be modelled by

$C\left(x\right)=6x^3-10x^2+10x+4$

Find the cost to the company on a day when no cricket bats are produced.

The company's daily revenue, $R$, in hundreds of AUD, from selling $x$ hundred cricket bats is given by $R\left(x\right)=42x$.

Given that profit = revenue - cost, find an expression for the daily profit, $P\left(x\right)$, in hundreds of AUD, from selling $x$ hundred cricket bats.

Find $P\text{'}\left(x\right)$.

The derivative of $P\left(x\right)$ gives the marginal profit. Production is at its profit-maximising level when the marginal profit is zero and $P\left(x\right)$ is positive.

Find the profit-maximising production level and the expected profit.

Dora makes an open container for collecting strawberries from a rectangular piece of cardboard measuring 55 cm by 28 cm. She cuts a square of side $x$ cm from each corner, as shown in the following diagram, and folds up the sides.

Show that the volume, $V$ cm³, of the container is given by

$V=4x^3-166x^2+1540x$

Find $\frac{\mathrm{d}V}{\mathrm{d}x}$.

Find

(i) the value of $x$ that maximises the volume of the container

(ii) the maximum volume of the container. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

A rice farm sells $x$ kg of rice each week. The weekly profit, $P$ dollars, from the sale of $x$ kg of rice satisfies

$\frac{\mathrm{d}P}{\mathrm{d}x}=-0.02x+6$, for $x\ge 0$.

Find the amount of rice, in kg, that should be sold each week to maximise the profit.

The profit from selling 250 kg of rice in a week is 480 dollars.

Find $P\left(x\right)$.

A paint company sells $x$ hundred litres of paint each week. The weekly profit, $P$ euros (€), from the sale of $x$ hundred litres of paint satisfies

$\frac{\mathrm{d}P}{\mathrm{d}x}=-1.9x+145$, for $x\ge 0$.

Find the number of litres of paint that should be sold each week to maximise the profit.

The profit from selling 7000 litres of paint in a week is €5000.

Find $P\left(x\right)$.

A skier performs a jump in a competition. She leaves the ground at point $\mathrm{Q}$ and travels in a vertical plane through $\mathrm{Q}$, landing at point $\mathrm{R}$, as shown in the following diagram.

Her path can be modelled by

$h\left(x\right)=-\frac{x}{80}\left(x-35\right)+30$

where $x$ metres is her horizontal displacement from $\mathrm{Q}$ and $h$ metres is her vertical displacement above $\mathrm{R}$. The model is valid from $\mathrm{Q}$ until she lands at $\mathrm{R}$.

Write down the vertical distance between $\mathrm{Q}$ and $\mathrm{R}$.

Show that the skier is again at the height of $\mathrm{Q}$ when $x=35$.

Find $h\text{'}\left(x\right)$. Hence find the total vertical distance travelled by the skier from $\mathrm{Q}$ to $\mathrm{R}$.

Percy has a sheet of metal 24 cm long and 33 cm wide. He has been asked to use it to make an open box to hold stationery. He removes a square of side $x$ cm from each corner of the sheet and then folds up the sides to make the box, as shown in the following diagrams.

Write down the possible values of $x$.

Show that the volume of the box, $V$ cm³, can be expressed as

$V\left(x\right)=4x^3-114x^2+792x$

Find $V\text{'}\left(x\right)$.

Using your answer to part (c), find the value of $x$ that maximises the volume of the box.

An open cylinder, with a base but no lid, has height $h$ cm and radius $r$ cm, as shown in the following diagram.

The sum of the diameter and the height of the cylinder is 20 cm.

Find an expression for the total surface area, $A$ cm², of the cylinder in terms of $r$.

Find $\frac{dA}{dr}$.

Find the value of $r$ for which the surface area is a maximum.

Two numbers, $x$ and $y$, are such that $x>y$ and the difference between them is 7.

Find the minimum possible value of the product $xy$, and the values of $x$ and $y$ that give this minimum.

A manufacturing company produces tins that must have a capacity of 470 cm³. Each tin is a closed cylinder with a height of $h$ cm and a base radius of $r$ cm.

Show that the surface area of a tin, $A$ cm², including the two circular ends, is given by

$A=2\pi r^2+\frac{940}{r}$

Sketch the graph of $A$ against $r$, for $r>0$.

The company would like to minimise the amount of metal used to make the tins.

(i) Find the coordinates of the stationary point on the graph of $A$, and justify that it is a minimum point.

(ii) Hence write down the minimum possible surface area of a tin, and the base radius that gives this minimum.

A tin of chopped tomatoes on sale in the UK has a capacity of 470 cm³ and a base radius of 3.7 cm.

Find the percentage difference between the surface area of this tin and the minimum possible surface area of a tin with the same capacity.

A chocolate manufacturer wants to introduce a new chocolate product in the shape of an equilateral triangular prism. Each edge of the triangular faces has length $x$ cm and the height of the prism is $h$ cm, as shown in the following diagram. The sum of the height of the prism and the three edges of one triangular face is 18 cm.

Explain why $x\ne 6$.

Show that the volume of the prism, $V$ cm³, can be written as

$V=\frac{3\sqrt{3}}{2}\left(3x^2-\frac{1}{2}{x}^3\right)$

By differentiating the expression in part (b), find the value of $x$ for which the volume is a maximum.

The chocolate is wrapped in packaging that exactly covers the surface of the prism.

Find the area of packaging needed when the volume of the prism is a maximum.

Patroclus, a would-be Olympic javelin thrower, throws a javelin during a training session. The height of the point of the javelin above the ground, $h$ metres, $t$ seconds after it is released can be modelled by

$h\left(t\right)=1.75+20.2t-4.90t^2$

Find $h\text{'}\left(t\right)$.

(i) Find the coordinates of the stationary point of the graph of $h$.

(ii) Justify that this stationary point is a maximum.

Find the greatest vertical distance that the point of the javelin travels above the height from which it was released.

A newly commissioned attack submarine is carrying out a series of manoeuvres to test its propulsion and steering systems. Its vertical position, $h$ metres, relative to sea level ($h=0$), $t$ minutes after it begins the manoeuvres, is given by

$h\left(t\right)=0.0125t^3-1.03t^2+16.6t-165$, for $0\le t\le 60$.

Find the coordinates of the stationary points of the graph of $h$.

For each of the stationary points found in part (a), determine whether it is a maximum point or a minimum point. Justify your answers.

Explain why, to find the maximum and minimum depths reached by the submarine for $0\le t\le 60$, it is not enough to consider only the stationary points found in part (a).

Find the greatest vertical distances that the submarine travels above and below the depth at which it started its manoeuvres, for $0\le t\le 60$.

A model rocket is launched vertically from the ground. The maximum height, $h$ metres, that it reaches can be modelled by

$h=\frac{2m^2}{15}-\frac{m^3}{480}$

where $m$ is the mass of fuel, in grams.

Find $\frac{\mathrm{d}h}{\mathrm{d}m}$, simplifying your answer.

Using your answer to part (a), find the mass of fuel that gives the greatest height, and state the greatest height.

The population of rabbits in a colony over one year can be modelled by the function

$f\left(t\right)=-\frac{1}{8}{t}^3+\frac{11}{5}{t}^2+20$, for $0\le t\le 12$,

where $t$ is the time, in months, from the start of the year.

Find the population of rabbits in the colony when $t=8$.

Find the maximum population of the colony and the time at which it occurs.

Find the value of $t$ at which the population is increasing most rapidly.

Daphne is starting a business selling fabric through an online retail website. She models the amount of fabric, $F$ metres, that she will sell in one month by

$F=151-5m$

where $m$ is the price per metre in US dollars (USD).

Daphne's business has fixed costs of 89 USD per month and variable costs of $3m$ USD per month.

Find an expression for the monthly profit, $P$ USD, in terms of $m$.

Find the price per metre, $m$, that gives Daphne the maximum monthly profit.

In her first month of trading, Daphne's costs must not exceed 128 USD.

Find the greatest profit that Daphne can make in her first month.

A petrol tanker transports fuel in a cylindrical container with radius $r$ metres and length $l$ metres, where $l=5r+3$. The container sits on top of a truck bed, and its diameter cannot exceed the width of the truck bed, 2.6 m.

Show that the volume, $V$ m³, of the container can be expressed as $V\left(l\right)=\frac{\pi l{\left(l-3\right)}^2}{25}$.

State the domain of the function $V\left(l\right)$.

Find the maximum volume of the container.

The transport company decides to use containers with the length found in part (c). Its costs, $C\left(x\right)$ dollars, depend on the size of the order and can be modelled by

$C\left(x\right)=\frac{7}{8}{\left(x-5\right)}^3-28x+390$

where $x$ is the volume of empty space in the container, in cubic metres.

Find the percentage of the container that should be filled for the costs to be a minimum.

Check, Mate! is a company that produces luxury chess sets for discerning chess set connoisseurs. The company's annual profit, $P\left(x\right)$ thousand pounds (£), can be modelled by

$P\left(x\right)=0.32x^3-12.4x^2+150x-480$

where $x$ is the number of chess sets sold per year, in hundreds. Because of manufacturing constraints, the company can sell at most 2500 chess sets in a year.

(i) State why there is no need to consider values of $x$ greater than 25.

(ii) Sketch the graph of $P\left(x\right)$ for $0\le x\le 25$.

(i) Find the coordinates of the stationary points of the graph, and state the number of chess sets sold and the profit that correspond to each point.

(ii) Find the maximum profit that the company can make in a year, and the number of chess sets that it must sell to make that profit.

Calculate

(i) the average rate of change of $P\left(x\right)$ between $x=5$ and $x=6$

(ii) the rate of change of $P\left(x\right)$ at $x=5$.

In each case, include the units and explain the meaning of the value you find.

State the values of $x$ for which the rate of change of $P\left(x\right)$ is negative. Explain the meaning of this result.

After analysing several years of data, a fast food company has found that the rate of change of its weekly sales can be modelled by

$\frac{\mathrm{d}M}{\mathrm{d}x}=-0.068x^3+0.72x^2-0.88x-1.9$, for $0\le x\le 10$,

where $M$ is the number of meals sold in a week, in thousands, and $x$ is the amount spent on advertising in the previous week, in thousands of euros.

In a week after 2000 euros was spent on advertising, 5988 meals were sold.

Find an expression for $M\left(x\right)$.

Find the maximum number of meals that the company can expect to sell in a week, and the amount that it should spend on advertising in the previous week to achieve this. Give your answers to the nearest meal and the nearest euro respectively. Justify that the value you find is a maximum.

Two numbers, $x$ and $y$, are such that $x>y$ and the difference between them is $k$, where $k$ is a positive constant.

Find the minimum possible value of $x^2+3y^2$, and the values of $x$ and $y$ that give this minimum. Give your answers in terms of $k$.

Justify that your answer to part (a) is a minimum value.

A perfume bottle is in the shape of a cone with base radius $r$ cm and height $h$ cm. All of its external surface is to be covered in gold leaf. The bottle must hold 95 cm³ of perfume, and the designer wishes to minimise the amount of gold leaf needed.

Show that the external surface area, $A$ cm², of the bottle can be written as

$A=\pi r\left(r+\sqrt{{\left(\frac{285}{\pi r^2}\right)}^2+r^2}\right)$

Find the radius of the bottle that minimises the surface area, and the minimum amount of gold leaf needed.

An artist is producing large pieces of sculpture for an art installation. Each piece is in the form of a cylinder with base radius $r$ metres, with a hemisphere of the same radius fitted exactly on top of the cylinder.

Every surface of each piece of sculpture, including the base, must be painted, so the artist wants a design for which, for any given volume, the total surface area is the minimum possible.

Show that for a piece of sculpture with volume $k\pi$ m³, the minimum surface area occurs when

$r=\sqrt[3]{\frac{3k}{5}}$

Find the minimum possible surface area for a piece of sculpture with volume $\frac{40}{3}\pi$ m³. Give your answer as an exact value.

An electric scooter travels between two cities at a constant speed of $x$ kilometres per hour. The energy used on the journey, $y$ kilowatt hours, can be modelled by

$y=\sqrt{8x}+\frac{25}{x}$, for $x>1$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ and hence find the speed that uses the least energy.

Calculate the least amount of energy that can be used for the journey.

A decorative bowl is made by casting metal into the shape of a hollow hemisphere, with internal radius $r$ cm and thickness $t$ cm, as shown in the following diagram.

A design constraint means that $r+2t=25$.

Find the maximum volume of metal that could be needed to make the bowl.

The manufacturer is not happy with the design when the bowl uses the maximum volume of metal. They claim that, with these dimensions, the volume of metal used to make the bowl is 80% more than the capacity of the bowl.

State whether the manufacturer's claim is correct. Justify your answer.

Muggins! is a company that produces luxury cribbage boards for discerning collectors of pub game paraphernalia. The company's monthly profit is $P\left(x\right)$ thousand pounds (£), where $x$ is the number of cribbage boards sold in the month, in hundreds.

For sales of between 0 and 100 cribbage boards in a month, the profit can be modelled by

$P\left(x\right)=4.53x^2-8.51$

For sales of between 100 and 1000 cribbage boards in a month, the profit can be modelled by

$P\left(x\right)=0.02x^3-\frac{9}{x}+5$

Because of manufacturing constraints, the company can sell at most 1000 cribbage boards in a month.

(i) Show that both formulae give the same profit for sales of 100 cribbage boards in a month.

(ii) State the values of $x$ for which each formula is valid.

On the same set of axes, sketch the graphs of the two profit functions. Sketch each function only over the values of $x$ for which it is valid.

Show that the combined profit function sketched in part (b) is increasing for all valid values of $x$ greater than zero.

Considering only values of $x$ for which $P\left(x\right)>0$, find the value of $x$ for which the rate of change of $P\left(x\right)$ is a minimum. Give this minimum rate of change, and explain its meaning in context.

Donty is a would-be social media celebrity who is obsessed with the number of likes his posts receive. He hires a statistician to study his social media accounts. After analysing several years of data, she finds that the rate of change of his number of likes can be modelled by

$\frac{\mathrm{d}L}{\mathrm{d}x}=-0.164x^3+2.73x^2-12.7x+15.3$, for $0\le x\le 12$,

where $L$ is the number of likes received on a given day, in thousands, and $x$ is the amount of new video content, in hours, that Donty uploaded on the previous day. Because of technical limitations, Donty cannot upload more than 12 hours of new video content in a day.

On a day after Donty uploaded 5 hours of new video content, he received 36 075 likes.

Find an expression for $L\left(x\right)$.

Find the maximum and the minimum number of likes that Donty can expect to receive in a day, and the corresponding number of hours of new video content that he should upload on the previous day. Justify that the values you find are the maximum and the minimum.

(i) For the maximum found in part (b), calculate the number of likes received for each minute of new video content uploaded on the previous day.

(ii) State, with a reason, whether the value found in part (c)(i) is the greatest number of likes per minute of new content that Donty can achieve.