5.5 Optimisation

A right-angled triangle of height , base and hypotenuse  has been rotated about its vertical axis to form a cone.

Write an expression for  in terms of .

Show that the volume of the cone, , can be expressed as:

Find the value of  which provides the cone with its maximum volume.

A wire of length 1 m is cut into two pieces. The first piece is bent into the shape of a square. The second piece is bent into a rectangle which has a length twice as long as its width  Let  cm be the perimeter of the square.

Find an expression for the area of the square.

Show that the width of the rectangle .

Find an expression for the sum of the area of the two shapes, 

Find the value of  such that the sum of the areas, , is a minimum.

Liam, a keen runner and swimmer, enters a competition which requires the competitors to run from point along a straight beach, before diving into the water and swimming directly to point . Liam is able to run at a speed of 8 m/s along the beach and swim at 2 m/s in the water.

Let  represent the distance between  and ,  the distance that Liam runs along the beach before entering the water and swimming along the line BC.

Find an expression for the time taken for Liam to run  metres between  and .

Show that the length of 

Find an expression for the total time taken for Liam to finish the race.

Find the value of  that will allow Liam to complete the race in the quickest time.

A small cylindrical drum, closed at the top but open at the bottom, has a radius  cm and a height cm. The volume of the drum is 

The material to make the top skin of the drum costs 25 cents per  and the curved surface of the drum costs 20 cents per .

Find an expression for  in terms of 

Show that the total cost of the material to make the drum is 

Find

The function  has a local minimum at the point .

Find the value of .

State, to the nearest dollar, the minimum cost required to make the drum.

Find  and hence, describe the concavity of the function  at .

The daily cost function of a company producing pairs of running shoes is modelled by the cubic function

where  is the number of pairs of running shoes produced and  the cost in USD.

Write down the daily cost to the company if no pairs of running shoes are produced.

The marginal cost of production is the cost of producing one additional unit. This can be approximated by the gradient of the cost function.

Find an expression for the marginal cost, , of producing pairs of running shoes.

Find the marginal cost of producing

pairs of running shoes

pairs of running shoes.

The optimum level of production is when marginal revenue, , equals marginal cost, . The marginal revenue, , is equal to 4.5.

Find the optimum level of production.

A cyclist riding over a hill can be modelled by the function

where  is the cyclist’s altitude above mean sea level, in metres, and  is the elapsed time, in seconds.

Calculate the cyclist’s altitude after a minute.

Find .

Calculate the cyclist’s maximum altitude and the time it takes to reach this altitude.

A company produces and sells cricket bats. The company’s daily cost, C, in hundreds of Australian dollars (AUD), changes based on the number of cricket bats they produce per day. The daily cost function of the company can be modelled by

where  hundred cricket bats is the number of cricket bats produced on a particular day.

Find the cost to the company for any day zero cricket bats are produced.

The company’s daily revenue, of AUD, from selling  hundred cricket bats is given by the function .

Given that profit = revenue - cost, determine a function for the profit,  in hundreds of AUD from selling  hundred cricket bats.

Find .

The derivative of  gives the marginal profit. The production of bats will reach its profit maximising level when the marginal profit equals zero and  is positive.

Find the profit maximising production level and the expected profit.

Dora decides to build a cardboard container for when she goes strawberry picking from a rectangular piece of cardboard, . She cuts squares with side length  cm from each corner as shown in the diagram below.

Show that the volume, , of the container is given by

Find 

Find

the value of that maximises the volume of the container

the maximum volume of the container. Give your answer in the form , where  and

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.