Trapezium rule

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.

Emma wants to know how much water is in a pond near her house.

She has taken width measurements of the pond every 3 m along its length. These width measurements are shown on the plan view of the pond below.

Emma has used these width measurements to sketch the following graph.

Emma knows that the pond has a uniform depth of 1∙2 m. Use the trapezium rule with 4 strips to calculate an estimate for the volume of water in the

pond.

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

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).

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.

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.

Alun is a jeweller. He is designing a symmetrical pendant, as shown below.

The pendant will be made from solid silver, with a uniform thickness of 3 mm. In order to calculate the cost of making the pendant, Alun wants to calculate an estimate of the volume of the pendant. He has accurately drawn one of the symmetrical halves of the shape on graph paper.

Calculate an estimate of the volume of the whole pendant.

Use the graph opposite, with 6 strips of equal width.

Alun makes pendants that are mathematical shapes.

At the end of last month, Alun took a stratified sample of 30 of these 240 pendants to check their quality.

Calculate how many pendants of each shape were in Alun’s sample.

You must show all your working.

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.

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.

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.