Graphical Methods (AQA Level 3 Mathematical Studies (Core Maths): Paper 2C: Graphical Techniques): Exam Questions

The sales of two chocolate bars (A and B) for the period 2000 to 2014 are shown.

A marketing executive says that from 2000–2010 bar A sales more than doubled. Is she correct?

Justify your answer.

Both chocolate bars are produced by the same company.

In 2014 the company decides to invest in a marketing campaign for one of the chocolate bars.

Use the evidence to advise the company which chocolate bar they should invest in for their marketing campaign.

Give reasons to justify your answer.

The market share of a web browser may be modelled by the equation

where:

• is the percentage market share

• is the number of months after October 2008

• and and are constants.

Table of values of Percentage Market Share at Months after October 2008

Estimate the values and .

You may use the grid on the next page if you wish.

The owner of the web browser estimates that the income from advertising per month is £23 000 per percentage point of market share.

Estimate the income from advertising in October 2010

Why would this model not be valid for large values of T ?

A colony of bacteria initially contains 4000 bacteria.

A scientist wants to know how long it will take for the size of the colony to double.

The number of bacteria, , after hours is given by

On the axes below, sketch the graph of for

Show the coordinates of any points where the curve crosses an axis.

Work out the number of bacteria after 6 hours.

Work out how long it takes for the number of bacteria to double from its initial value of 4000

Alia says,

“It will always take the same amount of time for the size of the colony to double from one given value to a size that is twice that value.”

Is Alia correct? Justify your answer.

The height of ocean waves is affected by wind speed.

Wave height, metres, can be modelled against wind speed, miles per hour, using the equation

Complete the table and draw the graph of against .

The Beaufort scale is used to classify and describe the force of winds.

The table shows the wind speed interval for some wind classifications.

(i) Work out an interval that describes the height of a wave in metres expected to be produced by a strong gale.

...................................................................................

[3]

(ii) State what property of the graph indicates that small changes in wind speed have more effect on the wave height during a storm than during a gale.

[1]

Describe one limitation of the model for predicting wave heights.

80 local football teams are ranked.

Each team has a ranking value, , from 1 to 80, with = 1 for the best performing team.

In August 2022, the ranking of all teams was revised using a new points-based system.

For each team, the number of points, , was initially determined using the equation

where was the ranking value in July 2022

Choose the graph which represents this equation.

For each game that a team plays, the expectation that the team wins, , is determined by the difference in ranking between the two teams,

For a team with ranking value ,

= Opponent ranking value –

The graph shows how varies with

Team A’s ranking value is 12

Team B’s ranking value is 48

Use the graph to work out the value of for both teams when they play each other.

Team A   E = ......................................................

Team B   E = ......................................................

The graph below represents the points a team gains, , if they win a game, based on the expectation of the team to win,

This data can also be represented by the equation

Use the graph to work out the values of the constants and

Playground slides are designed so that children slow down safely at the end of the slide on a horizontal run-out.

The diagram shows the dimensions important in slide design.

is the height of the slide.
is the slope length of the slide.
is the run-out.

A slide is designed with slope length = 3m

On this slide a child has an average speed of 2.04ms^–1 down the slope.

When designing playground slides, certain safety conditions must be met.

The length of the slope determines the maximum height permitted.

This condition is represented by the shaded region (R) on the graph.

Which one of the following designs will not meet the safety conditions?

Tick (✓) one box.

A tennis ball is hit across a tennis court.

The graph shows the height of the ball as it travels across the court.

Choose the type of function that the graph represents.

A second ball is hit so that

• it has an initial height 10cm below the initial height of the first ball

• it descends constantly by 10cm for every metre travelled across the court.

By plotting the graph of the second ball, work out the distance across the court when both balls are at the same height.

Answer .......................................... metres

In front of Sheffield train station is a curved sculpture that has a circular cross section at the front end and an oval cross section at the back end.

Over the length, metres, of the sculpture its height, metres, increases and its width, metres, decreases.

The rate of change of both height and width is constant along the full length of the sculpture.

The table shows information about the sculpture.

Work out the height and width halfway along the sculpture.

height = ........................................................ m

width = ......................................................... m

(i) The following function can be used to model the perimeter, metres, of the cross section.

Complete the table and plot the graph that represents this model.

(ii) Use your graph to describe how the perimeter of the cross section changes from the front end to the back end.

[1]

A computer game has five levels.

The score, , a player achieves on the game is dependent on two variables,

the number of enemies defeated, ,

and

the level on which the game is played, .

The score is calculated using the formula

The maximum possible score on a level is achieved by defeating all of the enemies on that level.

The number of enemies increases by 50 each time the player moves up a level.

Complete the table.

In a tournament, players are awarded an adjusted score, , using the formula

Two tournament players have their scores recorded.

The table shows some of the adjusted scores for the two players.

(i) Complete the table, giving each value to two decimal places.

[2]

The graph shows the adjusted scores for Player One.

(ii) Plot the adjusted scores for Player Two on the same graph.

[1]

(iii) A sixth level is added to the game.

Player Two makes the following prediction.

“I predict I will get a higher score than Player One on level 6”

Use the graph to comment on Player Two’s prediction.

[2]

Nikita conducts an experiment to measure how the temperature of drinks can be lowered by adding ice to a cooler box.

She repeats the experiment for different masses of ice.

The drinks are all initially at 22 °C and she records the final temperature of the drinks when all of the ice has melted.

She enters these readings into a computer that produces a graph modelling the final temperature, °C, of the drinks for the mass of ice, kg, added to the cooler box.

An ideal temperature for cold drinks is 4 °C

Use the model to estimate the minimum mass of ice required to cool them to 4 °C

Answer ..................................kg

Nikita states that increasing the mass of ice from 0.2 kg to 0.3 kg has more of an effect than increasing the mass of ice from 2.0 kg to 2.1 kg

Is Nikita correct?

Use the graph to explain your answer.

You do not have to work out the rate of change.

The model can also be represented by the equation

Using the measurements from Nikita’s experiment, work out the values of and .

B = .................................C = ................................

Describe one limitation of this model.

You may refer to the graph or the equation.

A rollercoaster is constructed by joining sections of track.

Equations are used to design each section of track.

The side elevation of the first section of the rollercoaster is shown on the graph.

is the horizontal distance from the start of the section

is the height of the track above ground level

The graph is linear for

Work out the equation of the line for

Give your answer in the form where and are constants.

Answer ...........................................

The next section of rollercoaster track is a downhill section.

The side elevation of this downhill section is shown on the graph below.

Which equation represents this downhill section of track?

Rollercoaster riders experience vertical g-force.

When vertical g-force is greater than 1, riders feel heavier than normal.

The graph below shows the vertical g-force that rollercoaster riders experience over a 70 m section of the track, where metres is the horizontal distance from the start of the section.

Estimate the time in seconds the rider will feel heavier than normal during this 70 m section.

Assume that the rollercoaster travels at a constant horizontal speed of 12 ms^–1

Answer.....................................................s

River levels in the UK are measured by a network of monitoring stations.

The chart below shows the data from one station on the river Don from 1 November 2019 to 1 March 2020

This section of river floods when the water level exceeds 4.8 metres.

Estimate the percentage of time during this 4-month period that the river flooded.

Answer........................................%

The graph below models the water level, metres, for a period of time, t hours, before and after 12.00 noon on 9 November 2019

For , which of the following types of function represents the data?

Use the graph above to work out the date and time of the maximum water level in this 192-hour period.

Date ...................................................Time ...........................................

After 12.00 noon on 9 November 2019, the water level, , is modelled by the equation

where and are constants.

(i) Use the graph to work out estimates for the values of and

=........................................... = ................................................

[3]

(ii) The water level is classed as ‘normal’ when below 3.18 metres.

Use the equation to forecast on what date after 9 November 2019 the water level would have dropped back below 3.18 metres.

You must show your working.

Answer ..................................

[3]

A domestic oil tank of height 140 cm has an octagonal cross-section with two square hollow sections of height cm

It takes 128 days to use all the oil when it is used at a constant rate.

The graph shows the oil level for the 128 days.

Use the graph to work out the height of the hollow sections, marked on the diagram.

Answer ........................................cm

A different tank is a cuboid of height 100 cm

During one week the rate of oil use increases at a steady rate.

The table shows oil level readings for the week.

(i) On the grid below, plot a graph modelling this data.

[2]

(ii) Use your graph to estimate the instantaneous rate at which the oil level is falling after 4 days.

State the units of your answer.

Answer .........................................................

(iii) Sketch a graph of the oil level in the tank if the rate at which the oil is being used decreases constantly over time.

[1]

A stone is dropped from a bridge into a river.

The distance, metres, that the stone has fallen after seconds is modelled by

Complete the table below.

Draw a graph of against on the grid below.

The stone hits the water when it has fallen 15 metres.

Use your graph to estimate the speed at which the stone hits the water.

Answer ..................................... m s^–1

Calculate the average speed of the stone from the moment it is dropped until it hits the water.

Answer ..................................... m s^–1

The Highway Code recommends that drivers on motorways allow at least a two-second gap between their vehicle and the vehicle in front.

Work out the recommended distance, in metres, between two vehicles which are both moving at the maximum legal speed of 112 km/h

Answer .......................... metres

Research suggests that having variable speed limits on motorways can maximise the flow of traffic.

Traffic on motorways can be modelled using

• density, vehicles per kilometre

• flow, vehicles per hour

• speed of traffic, kilometres per hour.

The graph shows a simplified relationship between and .

(i) Explain what is happening when the density is 175 vehicles per kilometre.

[1]

(ii) Traffic speed and density follow the model

where and are constants.

Use the graph to work out the value of and the value of .

= .............................................

= .............................................

[3]

(iii) Interpret your value of .

[1]

(iv) Flow, , and density, , are connected by the formula

where and are the values you worked out in question 6(b)(ii).

This model is represented in the graph below

Calculate the optimal flow as suggested by this model.

Answer ...............................vehicles per hour

[3]

(v) Work out the speed of traffic for optimal flow.

Answer.................................km/h

[2]

A boat is travelling at a constant speed on a straight canal.

The boat travels 150 metres in 60 seconds.

Calculate the speed of the boat.

Answer .......................................m s^–1

A child is cycling on a path next to the canal.

The child sets off when she is level with the front of the boat.

The graph shows how the distance travelled by the child and the distance travelled by the boat vary with time, seconds.

State all the values of t when the child is level with the front of the boat.

Answer......................................

Use the graph to estimate the speed of the child when

Answer ....................................... m s^–1

The distance, metres, travelled by the child at time seconds is modelled by

Work out the value of .

Answer ..................................

Extreme sports, including wingsuit flying, are rapidly increasing in popularity.

Wingsuit flying involves skydiving from a high altitude whilst wearing a special wingsuit which allows flyers to glide, thereby travelling horizontally as well as vertically.

In one competition, flyers are awarded points for their average horizontal speed during the competition window. The competition window is a time interval which starts when the flyer is at an altitude of 2960 metres and ends when the flyer is at an altitude of 1980 metres.

The table below shows information about one flyer during this competition.

For this flyer, show that the average horizontal speed, v_H, during the competition window is 47.54 m s^–1, correct to 2 decimal places.

The graph below shows the horizontal speed of the flyer during the competition window.

Bonus points are awarded if the flyer exceeds their average horizontal speed for at least 25% of their competition window.

Was this flyer awarded bonus points?

Show working to justify your answer

How many times during the competition window was the flyer moving with zero horizontal acceleration?

Give a reason for your answer.

A company that makes candles has just launched a new product.

The new candles have:

• a lower part which is a cylinder of diameter 6 cm and height 5 cm

• an upper part which is a truncated cone that narrows to a diameter of 2 cm

• a total height of 11 cm

The diagram shows a cross section of the candle.

The graph shows how, when the candle is burning, its height, cm, varies with time, hours.

Describe the rate of change of the height both before hours and after hours.

After the upper part of the candle has burned, the height, cm, follows the model

for

where is the time, in hours, for which the candle has been burning.

Work out the values of , and , given that:

• every hour, the height of the candle decreases by 0.39 cm

• the candle burns out when = 19.2 hours.

= .............................

= ...............................

= ................................

The company wants to change their design so that the candle will burn for exactly 24 hours.

The new candle has a taller lower part with the same diameter as before. The upper part does not change.

Both parts of the candle burn at the same rate as before.

Calculate the height of the new candle.

Answer .............................. cm