Further Modelling with Functions (DP IB Maths: AI HL)

Ben is planning an event, so he decides to investigate potential catering companies. Company A quotes a price of $1900 and says they will provide staff for food and drink service at an extra cost of $200 per hour, for a minimum of two hours. Company B quotes a price of $2100 and says they will provide staff for food and drink service for $160 per hour for a minimum of one hour and a maximum of six hours. 

Write down two functions, including their domains, for the total revenue generated by company A and company B from catering the event.

Ben has already calculated other costs involving in running the event and has concluded that the event needs to be at least three hours long in order to make a profit.

On the same diagram, sketch the graphs of both of the functions found in part (a).

Determine which catering company Ben should use to minimise the cost based on the duration of the event.

A Ferris-wheel-type attraction at a wildlife park is partially submerged underwater to enable passengers to observe both land and aquatic creatures at the park. The wheel rotates at a constant speed and the height of a viewing capsule above the water level is modelled by the function

where is the height, in metres, of the viewing capsule, and t is the elapsed time, in minutes, since the start of the ride. 

Find 

          (i)     the height above the water level at which passengers enter a viewing capsule, 

          (ii)    the maximum height above the water level that a viewing capsule reaches, 

          (iii)   the maximum depth below the water level that a viewing capsule reaches, 

          (iv)   the amount of time spent on one full rotation on the Ferris-wheel.

Calculate, to one decimal place, the length of time for which a capsule is under water during one ride.

The sound intensity level L, measured in decibels (dB), is given by , where I is the sound intensity and  is the reference sound intensity. 

Write down the sound intensity level, dB, in the case where  .

A manufacturing process takes place inside a sealed chamber and produces a pollutant that decays over time.  After the process is completed, at time t = 0 seconds, the amount of pollutant, P  ppm (parts per million) in the chamber is modelled by

Write down an expression for

The half-life of the pollutant is .

Find the value of .

It is safe for the chamber to be opened once the amount of pollutant falls to an amount

that is less than 10% above the background level.  

Given that , find the minimum number of complete minutes the chamber should remain sealed for after the manufacturing process completes.

Once it has reached a certain height above ground level, the height, h centimetres, of a particular species of sunflower can be modelled by the logistic function

where  is the number of days after the model becomes applicable and  is a constant.

Given that

According to horticulturists, this species of sunflower takes 80 – 120 days to reach its full height. Give one limitation of the model.

A model of the form , where , is proposed by an internet search engine to estimate the number of visits per day to websites. 

Each website is given a rank, , where , based on how popular the search engine company decides that website is.

Write down, in terms of , the lowest and highest possible visits a website can achieve in a single day according to the model.

Sketch the graph of  against . Label the points corresponding to the answers found in part (a).

Given that the graph of  against  passes through the point , find the value of . Give your answer correct to one significant figure.

Show that a website with a rank of 7 would have approximately 80 times as many visits per day as a website with a rank of 3 would have.

The water depth, d m (metres), at a port can be modelled by the function

where t is the elapsed time in hours since midnight. A and B are constants. 

With regards to the function , and giving your answers in terms of  and/or as appropriate, write down 

          (i)     the phase shift, 

          (ii)    the period, 

          (iii)   the amplitude, 

          (iv)   the equation of the principal axis.

Explain what the period and the amplitude of  mean in terms of the depth of water in the port.

Find the time of day at which the water depth first reaches its minimum value.

All water vehicles are prohibited from entering the port whilst the depth of water is below 3 m.

It is given that  and .

Find the times of day between which all water vehicles are prohibited from entering the port.

The sketch below shows the graph of  against .
The graph passes through the point (0,4) and has a gradient of 1.5.

Write down a function for , in terms of .

The sketch below shows the graph of  against .

The graph passes through the point  and has a gradient of 0.5.

Write down a function for  in the form .

The profile of a non-symmetrical skate ramp is modelled by the piecewise function

where m is the height of the ramp above ground level at the point  from the southernmost end of the skate ramp. 

Verify that both the southernmost and northernmost ends of the skate ramp are at a height of 2.5 m above ground level.

Find the height of the ramp when it is a distance of  from the southernmost end of the skate ramp.

Find the other distance from the southernmost end of the skate ramp when the ramp is at the same height as the answer to part (b). Give your answer to a sensible degree of accuracy.

Briefly explain, why, in this context, negative values of  could be considered.

The magnitude of an earthquake, , on the Richter scale can be modelled by the function

where E is the amount of energy, in joules (J), released by the earthquake. 

Find the magnitude, to one decimal place, of an earthquake which releases J of energy.

In February of 2011 an earthquake with magnitude 6.3 struck the region of Canterbury in the South Island of New Zealand.

Find the amount of energy, in joules, released by this earthquake, giving your answer in standard index form, correct to 2 significant figures.

The average time,T  in minutes, it takes for people to choose a movie to watch at the cinema can be modelled by the function

where  is the number of movies being shown at the cinema.

Calculate the number of minutes it would take someone to choose a movie at a cinema that is playing seven movies.

Find an expression for the inverse function .

It takes 1.073 minutes for someone to choose a movie at a cinema. Find the number of movies being shown at the cinema.

If a substance decays by per year, its half-life, H years, is given by

Find the half-life of a substance with an annual percentage decay rate of 12.2%.

Find the annual percentage decay rate of a substance with a half-life of 32.5 years.

Americium-241 has a half-life of 432.2 years and a sample initially contains 40 grams.

Find the number of grams in the sample after 55 years.

Consider the function , where  is a function of  and

Sketches of the graphs of  against  and  against , are shown below.

Use the information given in the graph sketches to find  in terms of .

The number of calculations, , a supercomputer can perform per second is related to the speed of its processor, s GHz.

The graph of  against  is an exponential curve passing through the point .

The semi-log graph of  against  is a straight line passing through the point .

Find , as a function of .

Kerry is organising her local town’s summer festival and wants to hire a bouncy castle for a minimum of 2 days up to a maximum of 12 days, depending on cost.  She has two potential suppliers to choose from.

Bounce-o-rama can supply a bouncy castle at per day for the first three days, per day for the next three days and £60 for every day thereafter.

Inflate-o-castle can supply a bouncy castle at per day, determined by the equation , where  is the number of days the bouncy castle is hired for, with a limit of 10 days. 

Given that Kerry has a budget of between £300 and £750, analyse which supplier she should use across the range of her budget in order to obtain the best value for money

State the number of whole days for which both suppliers charge the same amount.

Briefly explain why it would not be in Inflate-o-castle’s interest to offer more than 10 days of bouncy castle hire using the given cost equation.

A company makes a device that can be attached to small items such that if the item were to be dropped into water, the item would float rather than sink.  In testing the device, an employee throws the device over the edge of a boat and the height, , of the device above the level of calm water  seconds after being thrown is modelled by the function

where is a constant. 

Given that the employee throws the keys from a height of  above the level of calm water, find the value of A.

Find 

          (i)     the maximum height above the calm water level the device reaches,

          (ii)    the time at which the device first reaches calm water level,

          (iii)    the maximum depth below the calm water level the device reaches.

Find the times between which the device is deeper than 0.5 m below calm water level.

Describe the motion of the device for large values of .

A solid substance, with an extremely high melting point temperature, has a volume of at .

The substance expands as it is heated, but has a carrying capacity of .  The volume of the substance is 0.0001 m³ when heated to  

The volume,  at temperature , of the substance is modelled by the function

where  and are positive constants.

Find the values of  and , giving the value of  to two significant figures.

Find the volume, to the nearest whole cubic centimetre, of the substance when it is heated to .

Briefly explain the relevance of the substance’s extremely high melting point temperature in the context of the question.

The value, , measured in thousands of dollars, of a mobile phone initially worth $1000, is modelled by the function

where the age of the mobile phone is y years and and c are constants. 

Write down the value of .

Find the age of the phone when its value has halved from its initial worth.

State an assumption the model makes about the mobile phone.

In a fairground “lift and drop” ride, a row of seated passengers are projected vertically upwards until the ride reaches its maximum height (“the lift”), then the ride returns to its starting position (“the drop”).

The height of a passenger, cm, above ground level at time t seconds after the ride begins is modelled by the function

where and  are constants. 

Given that 

• passengers are 0.75 m above the ground when they board the ride, 

• the maximum height a passenger reaches on the ride is 12.25 m, 

• it takes 24 seconds for the ride to complete one “lift and drop”, 

• c takes the smallest positive value possible, 

find the values of  and .

A temporary lamppost that lights part of a forest pathway is powered via an outdoor electricity generator.  The cost,  of running the generator for  hours is modelled by the equation

where  and  are positive constants.

Given that, to three significant figures, the cost of running the generator for 2.31 hours is the half the cost of running the generator for 7.28 hours, find the value of b correct to one decimal place.

Given that  is an integer and that £1 would run the generator for just under 20 minutes, find the value of .

Find 

          (i)     the cost of running the generator for 12 hours, to the nearest penny (£0.01), 

          (ii)    the maximum length of time (in hours and minutes) the generator can be (continuously) run for at a maximum cost of £8.

A model of the form  is proposed to model the amount of a toxic gas, P parts per million (ppm), in a chamber, t seconds after a chemical reaction has taken place inside the chamber. and  are positive constants.

Write down, in terms of , the exact half-life, of the toxic gas.

Given that the amount of toxic gas reduces by 15% every 20 seconds, find its half-life to the nearest second.

The chemical reaction also produces a harmless gas which has some background level inside the chamber.  The amount of this gas increases with time after the chemical reaction has taken place.

Suggest a model for the amount of harmless gas in the chamber seconds after the  chemical reaction completes, defining any constants used and any restrictions on their values.

The Theoretical Zipf Distribution relates the rank and frequency of words used in the English language. 

For a large body of text a word of rank , appears  times where  ,  is the number of times the most used word occurs.

The graph below shows  plotted against for a particular novel in which the most used word is “the” and occurs 5623 times.

The graph passes through the points  and .

Use the graph to find  in terms of  and comment on whether the distribution of words in this novel follows the Theoretical Zipf Distribution.

The 8^th ranked word in the novel was “was”.  Use your answer to part (a) to estimate how many times “was” was used in the novel.

The profile of a mountain is modelled by the piecewise function

where km is the height of the mountain at a horizontal distance km along the ground from its basecamp.

is a continuous function at the point .

Find the value of  and the height of the mountain at this point.

Find

Find the height and the horizontal distance from mountain’s basecamp at the point where the slope of the mountain is flat.