Optimisation (DP IB Maths: AA HL)

An open cylinder with height h and radius r is shown in the diagram below.

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

Write down an equation for the total surface area of the cylinder, A, in terms of r.

Find .

Find the value of r when the surface area is maximised.

The population of rabbits in a colony is modelled over the course of a year by the function

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

Find the population of rabbits in the colony at .

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

Find the value of t for which the population of rabbits is increasing most rapidly.

Daphne is starting a business selling fabric via an online retail website. The amount of fabric, in metres, that she thinks she will sell in one month, F, can be modelled by

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

Daphne will incur various costs through running her business, both fixed and variable. The fixed costs total  per month and the variable costs can be expressed by 3m.

Write down an expression for the monthly profit, P, in terms of m.

Find the price per metre, m, that will give Daphne the maximum monthly profit.

In her first month of trading, Daphne must not let her costs exceed $128.

Find the greatest profit that Daphne is able to make in the first month.

Patroclus, a would-be Olympic javelin thrower, throws a javelin during a training session.  The height of the javelin’s point can be modelled by the equation

where  is the time, in seconds, that has passed since the javelin was released, and  is the height of the javelin above the ground, in metres.

Find .

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

A petrol tanker transports fuel in a cylindrical container with length  metres, where . The cylindrical container sits on top of a truck bed and cannot exceed its width of 2.6 m.

Show that the volume, V, of the container can be expressed as .

State the domain of the function .

Find the maximum volume of the container.

The transportation company decides to use containers with the length used in part (c). They find that their costs , in dollars, vary depending on the size of the order and can be modelled by the function

where is the number of cubic metres of empty space within the container.

d)     Find the percentage to which the container should be filled at which the costs are at a minimum.

Percy has been given a sheet of metal of length 24 cm and width 33 cm. He has been asked to use this material to design an open box that will be used to hold stationery. He removes a square of length  from each corner of the sheet of metal and the remainder of the sides are then folded up to create the box as shown in the diagrams below.

Write down the interval for the possible values of .

Show that the volume of the box, , can be expressed as

Find .

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

A chocolate manufacturer wishes to introduce a new chocolate product in the shape of an equilateral triangular prism. The edge of each triangular face has length  cm and the height of the prism is  cm, as shown in the diagram below. The sum of the height of the prism and all three triangular edges is 18 cm.

Explain why .

Show that the volume of the triangular prism, , can be described by

By differentiating the equation in part (b), find the length of  for which the volume is a maximum.

Find the amount of packaging required when the volume of the box is at a maximum.

Check, Mate! is a company that produces luxury chess sets for discerning chess set connoisseurs. The company’s profits , in thousands of UK pounds (£1000), can be modelled by the function

where  is the number of chess sets (in hundreds) sold per year.  Because of manufacturing constraints, the maximum number of chess sets that the company can sell in a year is 2500.

State why there is no need to consider values of  greater than 25.

Sketch a graph of  for  .

Calculate

the average rate of change of between  and .

the instantaneous rate of change of at .

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

State the values of  for which the instantaneous rate of change of is negative. Explain the meaning of this result.

A manufacturing company is producing tins that must have a capacity of 470 . The tins are in the shape of a cylinder with a height of  cm and a base radius of  cm.

Show that the surface area of the cylinder in , including the two circular ends, may be written as

Sketch the graph of  .

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

Find the stationary point on the graph of ,  and justify that it is a minimum point.

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

Determine the percentage difference between the surface area of that tin of chopped tomatoes and the minimum possible surface area for a tin with the same capacity.

Two numbers,  and , are such that  and the difference between the two numbers is 7.

Find the minimum possible value of the product , and the values of  and  that correspond to that minimum value.

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

where  represents the number of meals sold in a week (in thousands of meals sold), and  represents the amount spent on advertising during the preceding week (in thousands of euros).

It is known as well that 5988 meals are sold in a week where 2000 euros had been spent on advertising during the preceding week.

Find an expression for .

Find the maximum number of meals that the company can expect to sell in a week, and the amount of money that the company should spend on advertising during the preceding week to bring about that level of sales. Give your answers to the nearest meal sold and the nearest euro, respectively.  Be sure to justify that the value you find is indeed a maximum.

A model rocket is released from the ground on a vertical trajectory. The height, , in metres that it can reach is modelled by

where  is the mass of fuel in grams.

Find   and simplify it.

Using your answer to part (a), find the mass of fuel that is required to reach the maximum height and state what the maximum height is.

An electric scooter travels between two cities at a speed of  kilometres per hour.  Its fuel consumption can be expressed by

 for  

where  is the number of kilowatt hours used.

Find  and hence the speed that uses the minimum number of kilowatt hours.

Calculate the minimum amount of electricity that will be consumed for the journey.

A cone shaped perfume bottle of radius, , and height,  is to have all of its external surface area covered in gold leaf. The bottle is required to hold  of perfume, but the designer wishes to minimise the amount of gold leaf required.

Show that the surface area of the bottle can be expressed as

Find the radius of the perfume bottle that will result in the minimum surface area and the minimum amount of gold leaf that is required.

A decorative bowl is to be created by casting metal in to the form of a hollow hemisphere, with internal radius  cm and thickness  cm, as shown in the diagram below.

A design constraint means that .

Find the maximum volume of metal that would be required in the construction of the bowl.

The manufacturer is not happy with the design when the bowl is created using the maximum volume of metal. They claim that, with these dimensions, the volume of metal used in the bowl’s construction is 80% more than the capacity of the bowl.

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

A particle P moves along a straight line. The velocity  ms^-1 of P after  seconds is given by

 for   .

The following diagram shows the graph of .

Find the initial velocity of P.

Find the maximum speed attained by P.

Write down the number of times that the acceleration of P is  ms^-2.

Find the acceleration of P when it changes direction the first time.

A newly-commissioned attack submarine is performing a series of manoeuvres to test its propulsion and steering systems.  The vertical position of the submarine relative to sea level (where sea level is represented by ) is given by the equation

where  is the time, in minutes, that has passed since the submarine began its manoeuvres, and  is the vertical position of the submarine in metres.

Find the stationary points for .

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

Explain why, in order to find the maximum and minimum depths reached by the submarine in the interval , it is not sufficient merely to consider the stationary points found in part (a).

Find the greatest vertical distances that the submarine travels in the interval  above and below the depth from which it started its manoeuvres.

Muggins! is a company that produces luxury cribbage boards for discerning collectors of pub game paraphernalia. For sales of between 0 and 100 cribbage boards in a month, the company’s profits , in thousands of UK pounds (£1000), can be modelled by the function

where  is the number of cribbage boards (in hundreds) sold during the month.  For sales of between 100 and 1000 cribbage boards in a month, the corresponding formula is

Because of manufacturing constraints, the maximum number of cribbage boards that the company can sell in a month is 1000.

State the ranges of  values for which each formula is valid.

On the same set of axes, sketch the two profit functions. Each function should only be sketched over the interval of  values for which it is valid.

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

Considering only values of  for which ,  find the value of  for which the instantaneous rate of change of  is a minimum.  Give the value of the corresponding instantaneous rate of change, and explain the meaning of that value in context.

An artist is producing large pieces of sculpture for an art installation. Each piece is in the form of a cylinder with base radius metres, on top of which is a hemisphere with the same radius as the cylinder’s base radius. The hemisphere is fitted exactly to the top of the cylinder, so that the circular bottom of the hemisphere lines up exactly with the circular top of the cylinder. 

Every side of each piece of sculpture must be painted, so the artist is eager to find a design for his sculptures such that, for any given volume of a piece of sculpture, the total surface area will be the minimum possible.

Show that for a piece of sculpture with volume , the minimum surface area occurs when

Find the minimum possible surface area for a piece of sculpture with volume  Give your answer as an exact value.

Two numbers,  and , are such that  and the difference between the two numbers is , where  is a positive constant.

Find the minimum possible value of the sum , and the values of  and that correspond to that minimum value. Your answers should be given in terms of .

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

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, and after analysing several years of data she determines that the rate of change of his number of ‘likes’ can be modelled by the equation

where  represents the number of likes received on a given day (in thousands of likes), and represents the amount of new video content Donty uploaded on the preceding day (in hours). Because of technical limitations, Donty is unable to upload more than 12 hours of new video content on any given day.

It is known as well that 36075 ‘likes’ are received on a day after 5 hours of video content was uploaded the day before.

Find the maximum and minimum number of ‘likes’ that Donty can expect to receive in a day, and the corresponding number of hours of new video content that Donty should upload on the preceding day to attain that maximum or minimum. Be sure to justify that the values you find are indeed the maximum and minimum possible.