Real-life graphs

Furlongs and yards are both measures of length.

In the past, these measures were used in farming. The conversion graph below can be used to convert between furlongs and yards.

Complete each of the following statements.

330 yards is equal to_........ furlongs.

7 furlongs is equal to_........ yards.

1320 yards is equal to_........ furlongs.

Furlongs and yards are both measures of length.

In the past, these measures were used in farming. The conversion graph below can be used to convert between furlongs and yards.

Complete each of the following statements.

330 yards is equal to_........ furlongs.

7 furlongs is equal to_........ yards.

1320 yards is equal to_........ furlongs.

The map below shows part of the coastline and some islands off the coast of Gwynedd.

Write down the bearing of Aberdaron from Bardsey Island lighthouse.

Write down the bearing of Bardsey Island lighthouse from Ynys Gwylan-bach.

Huw can see Bardsey Island lighthouse from Ynys Gwylan-bach. How far is the lighthouse from Ynys Gwylan-bach? Give your answer in kilometres.

You must show all your working.

The candela and the lumen are units that can be used to measure light intensity.

Below is a conversion graph. You can use this graph to make approximate conversions between candelas and lumens for a particular type of light.

For this type of light, complete each of the following statements.

1·15 candelas is approximately equal to_........ lumens.

13·5 lumens is approximately equal to_........ candelas.

The light from Bardsey Island lighthouse has an intensity of approximately 52 000 candelas. The light from Strumble Head lighthouse in Pembrokeshire has an intensity of approximately 1 000 000 candelas.

By estimating, complete the following statement.

You must show all your working.

‘The light from Strumble Head lighthouse is approximately_........ times as intense as the light from Bardsey Island lighthouse.’

Yesterday, Jemila cycled along a straight track from home to the beach and back.

Distance from home (km)

For how many minutes was Jemila 17·5 km from home?

At what time did Jemila first start cycling in the direction of home?

Circle your answer.

13:30 14:15 15:00 15:30 16:00

By 15:30, how many kilometres in total had Jemila cycled?

Jemila was due to get home at 16:30. She arrived home early.

How many minutes early was she?

The map below shows part of the coastline and some islands off the coast of Gwynedd.

Write down the bearing of Aberdaron from Bardsey Island lighthouse.

Write down the bearing of Bardsey Island lighthouse from Ynys Gwylan-bach.

Huw can see Bardsey Island lighthouse from Ynys Gwylan-bach. How far is the lighthouse from Ynys Gwylan-bach? Give your answer in kilometres.

You must show all your working.

The candela and the lumen are units that can be used to measure light intensity.

Below is a conversion graph. You can use this graph to make approximate conversions between candelas and lumens for a particular type of light.

For this type of light, complete each of the following statements.

1·15 candelas is approximately equal to_........ lumens.

13·5 lumens is approximately equal to_........ candelas.

The light from Bardsey Island lighthouse has an intensity of approximately 52 000 candelas. The light from Strumble Head lighthouse in Pembrokeshire has an intensity of approximately 1 000 000 candelas.

By estimating, complete the following statement.

You must show all your working.

‘The light from Strumble Head lighthouse is approximately_........ times as intense as the light from Bardsey Island lighthouse.’

On Tuesday, Alfred travelled on a straight road.

The graph represents his journey during the day, until the time he arrived home.

Distance from home (km)

At what time did Alfred arrive home on Tuesday?

How far, in total, did Alfred travel during the day on Tuesday?

Sian ran a 100-metre race.

The graph below shows Sian’s speed over the first 8 seconds of the race. Sian then ran the rest of the race at a constant speed of 8 m/s.

Speed (m/s)

By first using the trapezium rule with 4 strips of equal width for the first 8 seconds, estimate the total time it took Sian to run the 100-metre race.

You must show all your working.

Sian’s total time to run the 100-metre race =_{……………........ …………} seconds

The graph below shows the depth of water in a reservoir during a 200-day period.

For approximately how many of the 200 days was the depth of water in the reservoir decreasing?

Circle your answer.

120 days 200 days 80 days 100 days 40 days

Use 5 strips of width 40 days to estimate the area under the curve. Hence, calculate an estimate of the average depth of the water during this 200-day

period.

Estimate the rate of increase in the depth of the water on the 60th day of this period.

Give your answer in its simplest form.

Distance along the canal from Llangollen (km)₁₂

During the day, Hesta and Walt made stops by the side of the canal.

At what time did Hesta and Walt first stop at the side of the canal?

What is the total distance Hesta and Walt travelled in the boat between 8 a.m. and 4 p.m.?

Circle your answer.

3 km 9 km 12 km 15 km 18 km

Hesta and Walt visit Chirk Castle.

Which is the best estimate for the bearing of Llangollen from Chirk Castle?

Circle your answer.

060° 240° 120° 340° 300°

Owain has drawn a conversion graph to help him with his homework.

Circle the correct answer to complete the following statement. 15°C in Fahrenheit is

–5°F 59°F 58°F 55°F 66°F

Bethan says,

Bethan is incorrect.

Express 40°C in Fahrenheit.

Distance along the canal from Llangollen (km)₁₂

During the day, Hesta and Walt made stops by the side of the canal.

At what time did Hesta and Walt first stop at the side of the canal?

What is the total distance Hesta and Walt travelled in the boat between 8 a.m. and 4 p.m.?

Circle your answer.

3 km 9 km 12 km 15 km 18 km

Hesta and Walt visit Chirk Castle.

Which is the best estimate for the bearing of Llangollen from Chirk Castle?

Circle your answer.

060° 240° 120° 340° 300°

The direct straight-line distance between Llangollen and the Pontcysyllte Aqueduct is 5. 6 km. On the map this distance is 5.6 cm. What is the scale of the map?

Circle your answer.

1: 10 1: 1000 1: 10 000 1: 100 000 1: 1 000 000

• It holds 1 500 000 litres of water. • It takes 2 hours to drain the water.

The section of the canal over the aqueduct is to be drained.

Calculate the number of litres of water that drain from this section of the canal per minute. You may assume that the water drains from the canal at a constant rate.

You must show all your working.

Calculate the area of the canal’s uniform cross-section.

Give your answer in cm².

Owain has drawn a conversion graph to help him with his homework.

Circle the correct answer to complete the following statement. 15°C in Fahrenheit is

–5°F 59°F 58°F 55°F 66°F

Bethan says,

Bethan is incorrect.

Express 40°C in Fahrenheit.

The travel graph below shows a journey Luke made on Saturday along a straight road.

How far away from home was Luke at 17:00?

For what length of time was Luke away from home on this journey?

Circle your answer.

During his journey, Luke visited a friend’s house. He stopped for an hour and then continued his journey.

How far from Luke’s home does his friend live?

OrenVit is a company that produces bottles of orange juice.

The company uses only bottles with a capacity of one litre. Each day, the cost of producing bottles of orange juice is as follows:

Draw a graph to show the total daily cost of producing between 0 and 100 bottles of orange juice.

Use the graph paper below.

One day, OrenVit produces 1750 pints of orange juice in one-litre bottles.

Calculate the cost of producing this quantity of orange juice.

The travel graph below shows a journey Luke made on Saturday along a straight road.

How far away from home was Luke at 17:00?

For what length of time was Luke away from home on this journey?

Circle your answer.

During his journey, Luke visited a friend’s house. He stopped for an hour and then continued his journey.

How far from Luke’s home does his friend live?

How many kilometres did Luke travel between 13:00 and 14:30?

OrenVit is a company that produces bottles of orange juice.

The company uses only bottles with a capacity of one litre. Each day, the cost of producing bottles of orange juice is as follows:

Draw a graph to show the total daily cost of producing between 0 and 100 bottles of orange juice.

Use the graph paper below.

One day, OrenVit produces 1750 pints of orange juice in one-litre bottles.

Calculate the cost of producing this quantity of orange juice.

Estimate the greatest acceleration that Eliska achieved during the first 8 seconds of her race.

Give your answer in its simplest form.

Use the trapezium rule with exactly 4 strips of equal width to estimate the distance

Eliska travelled during the first 8 seconds of the race.

The speed-time graph below shows Eliska’s speed from t = 8 seconds to t = 48 seconds. The graph shows that her speed increased at a constant rate from t = 8 to t = 16, and at a different constant rate from t = 16 to t = 48.

Speed (m/s)

Eliska’s speed at t = 48 was 1 m/s faster than her speed at t = 16. From t = 8 to t = 48, Eliska travelled 550 m.

Form and solve an equation to calculate Eliska’s speed at time t = 16 seconds.

Eliska’s speed at time t = 16 seconds is_{………........ ………………..} m/s

Dean went to the gym yesterday afternoon.

The graph shows the distance Dean was from home during yesterday afternoon.

Distance from home (km)

How far away from home was Dean at 15:15?

Circle your answer.

15∙5 km 15 km 16∙5 km 16 km 17 km

At what time did Dean arrive back home?

Circle your answer.

5:30 p.m. 5:30 a.m. 5:15 p.m. 5:10 p.m. 5:00 a.m.

A water tank is filled every morning.

The graph below shows the depth of water in the tank between 8:00 a.m. and 9:00 a.m. on Friday and Saturday.

Depth of water (mm)

What was the difference between the depth of water on Friday and on Saturday at

8:20 a.m.?

Consider the time interval between 8:10 a.m. and 8:50 a.m. At what time was the depth of the water in the tank the same on both Friday and

Saturday?

The tank can hold water to a depth of 400 mm.

On Saturday, at what time was the water in the tank half this depth?

8:28 a.m. 8:20 a.m. 8:35 a.m. 8:12 a.m. 8:30 a.m.

Dean went to the gym yesterday afternoon.

The graph shows the distance Dean was from home during yesterday afternoon.

Distance from home (km)

How far away from home was Dean at 15:15?

Circle your answer.

15∙5 km 15 km 16∙5 km 16 km 17 km

At what time did Dean arrive back home?

Circle your answer.

5:30 p.m. 5:30 a.m. 5:15 p.m. 5:10 p.m. 5:00 a.m.

A water tank is filled every morning.

The graph below shows the depth of water in the tank between 8:00 a.m. and 9:00 a.m. on Friday and Saturday.

Depth of water (mm)

What was the difference between the depth of water on Friday and on Saturday at

8:20 a.m.?

Consider the time interval between 8:10 a.m. and 8:50 a.m. At what time was the depth of the water in the tank the same on both Friday and

Saturday?

The tank can hold water to a depth of 400 mm.

On Saturday, at what time was the water in the tank half this depth?

8:28 a.m. 8:20 a.m. 8:35 a.m. 8:12 a.m. 8:30 a.m.

Dilwyn draws a conversion graph to help him understand his geography homework. He is looking at areas, in both square metres (m²) and acres.

Square metres (m²)

Which of the following is equivalent to 0.05 acres? Circle your answer. (a)

Which of the following is equivalent to 0.3 acres? Circle your answer. (b)

The graph below shows the velocity, v, in m/s, of a particle at time t seconds, for the first

8 seconds of its journey.

Velocity, v (m/s)

At time t = 2 seconds, estimate the acceleration of the particle. Simplify your answer as far as possible.

State the units of your answer.

Use the trapezium rule with 8 strips of equal width to estimate the distance travelled

by the particle in the first 8 seconds.

The actual distance travelled during the first 6 seconds is exactly 144 m. Use this to calculate the average velocity of the particle over the first 6 seconds

Give your answer in miles per hour (mph).

Meinir is planning a charity event to be held at a hotel.

A section of a straight line graph showing the hotel charges for this event is shown below. These charges include a single payment for the room hire and the cost of one drink for each person attending.

Overall cost (£)

Meinir decides to pay the room hire cost herself. She decides to price the tickets so that she will be able to make £500 to give to charity.

Calculate the selling price of each ticket if Meinir plans the event for 50 people.

Calculate the selling price of each ticket if Meinir plans the event for 400 people.

Meinir is planning a charity event to be held at a hotel.

A section of a straight line graph showing the hotel charges for this event is shown below. These charges include a single payment for the room hire and the cost of one drink for each person attending.

Overall cost (£)

Meinir decides to pay the room hire cost herself. She decides to price the tickets so that she will be able to make £500 to give to charity.

Calculate the selling price of each ticket if Meinir plans the event for 50 people.

Calculate the selling price of each ticket if Meinir plans the event for 400 people.

Carwyn owns a car and a motorcycle.

Write down a formula for the value V, in pounds, of Carwyn’s motorcycle after he

has owned the motorcycle for t years.

Which one of the following best represents the graph of V against t?

Circle your answer.

Eleri’s cycle computer shows a speed-time graph for the first 60 seconds of her cycle ride.

Speed (ms⁻¹)

The cycle computer has also generated the following table, showing Eleri’s speed after every 10 seconds, for the first 40 seconds of her ride.

Calculate an estimate of the distance travelled in the first 40 seconds of Eleri’s cycle ride.

You must use 4 strips of equal width.

23 23

After 40 seconds, Eleri’s speed remained constant.

Calculate an estimate of Eleri’s average speed over the first 60 seconds of her ride. Give your answer in ms⁻¹, correct to 2 significant figures.

You must show all your working.

Emily has drawn a conversion graph, as shown below.

She uses it to help her brother understand how to convert knots to miles per hour.

Miles per hour

Complete each of the following statements.

23 miles per hour is equal to_........ knots.

5 knots is equal to_........ miles per hour.

Emily has drawn a conversion graph, as shown below.

She uses it to help her brother understand how to convert knots to miles per hour.

Miles per hour

Complete each of the following statements.

23 miles per hour is equal to_........ knots.

5 knots is equal to_........ miles per hour.

The velocity-time graph below shows the velocity of an aircraft for the first 20 seconds of take-

Calculate an estimate of the acceleration of the aircraft at time 16 seconds.

Calculate an estimate of the distance travelled by the aircraft during the first 20 seconds of take-off. Use 5 strips of equal width.

You must convert your answer into miles. Give your answer as a decimal.

Distance is_{……………........ …………..} miles

The aircraft carries cargo. One customer wants to use the aircraft to transport a new product that is to be packaged in cuboid boxes, as shown below.

The boxes will meet the following conditions: • The boxes will be of width 8 cm and length 10 cm. • The length of the diagonal AB will be 14 cm.

Calculate the height of a box.

Aled and Gareth went on holiday to France.

The total cost of the holiday was £660. • Aled’s mother paid of the total cost. • Aled and Gareth shared the remaining cost in the ratio 1: 9.

Calculate how much each person paid towards the cost of the holiday.

Aled’s mother paid £_........

Explain how you could use your answers to check that they are correct.

Gareth’s luggage weighed 21.13 kg. This was over the maximum of 20 kg allowed.

Gareth removed items from his luggage so that its mass was: • as close to 20 kg as possible, • not greater than 20 kg.

From the following list of items, which two items did Gareth remove?

You must show all your working.

Before going on holiday, Aled made a conversion graph to help him understand prices in

Use Aled’s conversion graph to answer the following questions.

A camera costs £90.

How much is this in euros?

Camera costs_........ euros

A meal costs £25.

How much is this in euros?

Meal costs_........ euros

Gareth looked at exchange rates for buying euros. He recorded the exchange rates for the previous 60 days, as shown below.

£1 = b euros	Frequency
1.00 b 1.04	2
1.04 b 1.08	8
1.08 b 1.12	16
1.12 b 1.16	33
1.16 b 1.20	1

Gareth started to draw a frequency diagram to show this information.

Complete the frequency diagram.

Which is the modal group?

Circle your answer.

Aled and Gareth went on holiday to France.

The total cost of the holiday was £660. • Aled’s mother paid of the total cost. • Aled and Gareth shared the remaining cost in the ratio 1: 9.

Calculate how much each person paid towards the cost of the holiday.

Aled’s mother paid £_........

Explain how you could use your answers to check that they are correct.

Gareth’s luggage weighed 21.13 kg. This was over the maximum of 20 kg allowed.

Gareth removed items from his luggage so that its mass was: • as close to 20 kg as possible, • not greater than 20 kg.

From the following list of items, which two items did Gareth remove?

You must show all your working.

Before going on holiday, Aled made a conversion graph to help him understand prices in

Use Aled’s conversion graph to answer the following questions.

A camera costs £90.

How much is this in euros?

Camera costs_........ euros

A meal costs £25.

How much is this in euros?

Meal costs_........ euros

Gareth looked at exchange rates for buying euros. He recorded the exchange rates for the previous 60 days, as shown below.

£1 = b euros	Frequency
1.00 b 1.04	2
1.04 b 1.08	8
1.08 b 1.12	16
1.12 b 1.16	33
1.16 b 1.20	1

Gareth started to draw a frequency diagram to show this information.

Complete the frequency diagram.

Which is the modal group?

Circle your answer.

A train manufacturer has developed a new braking system. The velocity-time graph shows the velocity of a train from when the new brakes are applied until it comes to rest.

Velocity (ms⁻¹)

Estimate the train’s deceleration at time 60 seconds.

Calculate an estimate of the distance travelled by the train from the instant the brakes are applied until it comes to rest.

You must use exactly 4 strips of equal width.

Explain how you could use the graph to gain a more accurate estimate of the

distance travelled.

The base radius of the cone and the radius of the cylinder are both 9 cm. The height of the cylinder is four times the vertical height of the cone.

The bottle has been designed so that it has a capacity of 10 litres.

Calculate the total height of the bottle.

Weedkiller is poured into the bottle at a constant rate. The cone is full of weedkiller after 2 seconds. The bottle as a whole is full after 26 seconds. Using the axes below, sketch a graph of the height of weedkiller in the bottle during the

26 seconds it takes to fill.

Height of weedkiller (cm)

Maxim is doing a project on shipping.

He draws and uses his own conversion graph to convert between nautical miles and miles.

What is 50 nautical miles converted into miles?

Circle your answer.

.... 55 56 5 57 5 58 5 59 5

Complete the following statement.

800 nautical miles is equal to_{…………........ ………….} miles.

Maxim is doing a project on shipping.

He draws and uses his own conversion graph to convert between nautical miles and miles.

Complete the following statement.

800 nautical miles is equal to_{…………........ ………….} miles.

Carys has to write a report on the water levels of the River Tad.

She has recorded the depth of the water in the River Tad between 1 p.m. and 7 p.m. This is shown in her graph below.

What was the greatest recorded depth of water in the river?

Circle your answer.

26 cm 27 cm 46 cm 48 cm 50 cm

For what period of time was the depth of water in the river greater than 45 cm?

Circle your answer.

48 minutes 1 hour 1 hour 12 minutes

1 hour 24 minutes 1 hour 30 minutes

Gwen fills a 10-litre bucket with water from a tap.

She turns the tap until it is fully open. The bucket fills up with water, and when Gwen thinks it is close to being full, she slowly closes the tap. The bucket is full after 20 seconds. The graph below shows the volume of water in the bucket during the 20 seconds.

After how many seconds did Gwen start to close the tap?

Estimate at what rate water is entering the bucket at time t = 10 seconds.

Give your answer in litres per second.

When the tap is fully open, water flows out at 2 litres per second.

Express 2 litres per second in gallons per minute.

You must show all your working.

Can a fully open tap fill a 90-gallon tank in under 3 minutes?

You must show all your working.

The atmosphere above the Earth’s surface is divided into different layers.

The graph shows the average temperature at different heights above sea level. It also shows the positions of the different layers of the atmosphere.

Use the graph to answer the following questions.

What is the approximate temperature 40 kilometres above sea level?

Circle your answer.

–30˚C 10˚C –22˚C –18˚C 130˚C

What is the approximate difference between the least and the greatest temperature shown

in the troposphere ?

Why does the graph not pass through (0, 0)?

What is the increase in the daily running costs for each additional dog that is kept in the

kennels?

Freda also runs a dog kennels. The cost of keeping 20 dogs in her kennels for one day is £130. She knows that as the number of dogs increases, the overall cost increases at the same rate as in William’s kennels.

Display this information on the graph paper opposite.

Find the cost of keeping 30 dogs for one day in Freda’s kennels.

Why does the graph not pass through (0, 0)?

What is the increase in the daily running costs for each additional dog that is kept in the

kennels?

Freda also runs a dog kennels. The cost of keeping 20 dogs in her kennels for one day is £130. She knows that as the number of dogs increases, the overall cost increases at the same rate as in William’s kennels.

Display this information on the graph paper opposite.

Find the cost of keeping 30 dogs for one day in Freda’s kennels.

Two runners, Catrin and Delyth, start a race at the same time.

The velocity-time graph shows their velocities over the first 5 seconds of the race.

After the start of the race, what was the earliest time that Catrin’s acceleration was

0 m/s²?

Use the trapezium rule to calculate an estimate of the distance Catrin travelled in the first 5 seconds of the race. Use Catrin’s velocities at times t = 0, t = 1, t = 2, t = 3, t = 4 and t = 5.

You must show all your working.

Calculate an estimate of how far Catrin was ahead of Delyth after 5 seconds.

Explain why your answer to (c) (i) is an underestimate.

Eleri and Yvon are sisters.

They both live at Cwm Uchel. They do not go to the same school. The graph represents each of their journeys to school and back.

Key:

Distance from home (km)

At what time did Yvon arrive home from school?

Circle your answer.

14:45 15:15 15:30 15:45 16:00

Eleri cycles along a straight road to school and back. How far does she cycle when going to school and back in one day?

Circle your answer.

6 km 8 km 9 km 12 km 16 km

Martha looks at the graph and says,

‘The school Eleri attends is 2 km from Yvon’s school.’

Is this true?

Certainly true
Certainly false
Can’t tell

Give a reason for your answer.

Alun has made his own conversion graph to change knots to miles per hour.

Use Alun’s conversion graph to write 150 knots in miles per hour.

Nikita thinks Alun’s conversion graph may be inaccurate.

Nikita knows that 1000 knots is 1150.779 miles per hour, correct to 3 decimal places.

Convert 20 knots to miles per hour • using Alun’s conversion graph, and then • using Nikita’s values.

Calculate the difference, in miles per hour, between your answers. Give your answer correct to 2 decimal places.

You must show all your working.

Marcus is a farmer.

He has his own conversion graph to change between acres and square yards.

Complete each of the following statements.

3 acres is equal to_{………………........ ……………………….} square yards.

5.5 acres is equal to_{………………........ ……………………….} square yards.

The travel graph below shows a journey Gareth made yesterday.

Distance from home (km)

How far away from home was Gareth at 15:00?

Circle your answer.

0 km 2 km 4 km 6 km 8 km

At what time did Gareth arrive back home?

Circle your answer.

14:00 16:30 18:45 19:15 19:30

Sometime after 5p.m., Gareth headed for the supermarket. The supermarket was closed when he got there so he headed straight back home. At what time did Gareth arrive at the supermarket?

Circle your answer.

17:00 17:30 18:00 18:15 18:30 19:00

Gareth did not stop for the whole of the time between 15:00 and 15:30.

What could the travel graph tell you about his journey between these times?

Marcus is a farmer.

He has his own conversion graph to change between acres and square yards.

Complete each of the following statements.

3 acres is equal to_........ square yards.

Petra is organising a prom for her year group. The number of people attending the prom is likely to be between 20 and 80.

The cost of holding the prom at Hotel Afonwen would be as follows. • Hire of the room: £100 • Food: £15 per person • Welcome drink on arrival: £3 per person • Decorations: £2 per person

Petra decides to share all the costs equally between the people attending.

• Let £P be the price paid per person. • Let N be the number of people attending the prom.

Write a formula for P, in terms of N.

Hiring a larger room at Hotel Afonwen costs £200. The cost per person for food, welcome drinks and decorations remains the same.

If the total cost is £2240, how many people attend?

Petra is organising a prom for her year group. The number of people attending the prom is likely to be between 20 and 80.

The cost of holding the prom at Hotel Afonwen would be as follows. • Hire of the room: £100 • Food: £15 per person • Welcome drink on arrival: £3 per person • Decorations: £2 per person

Draw a graph to illustrate the total cost of holding the prom for between 20 and 80 people.

Use the graph paper below.

Petra decides to share all the costs equally between the people attending.

• Let £P be the price paid per person. • Let N be the number of people attending the prom.

Write a formula for P, in terms of N.

Hiring a larger room at Hotel Afonwen costs £200. The cost per person for food, welcome drinks and decorations remains the same.

If the total cost is £2240, how many people attend?

During which quarter-hour period was Siân’s acceleration the greatest?

At about what time did Siân stop accelerating?

Siân usually finds cycling at a velocity of 18 km/h very comfortable.

Express 18 km/h in metres per second.

Using her velocities at 14:00, 15:00 and 16:00, calculate an estimate for the total distance

Siân travelled between 14:00 and 16:00.

Distance travelled_{………........ ………} km

In this part of the question, you will be assessed on the quality of your organisation, communication and accuracy in writing. Siân estimated the distance she travelled between 16:00 and 17:00 as 5 miles. Is Siân’s estimate reasonable?

You must justify your answer and show your working.