Rates of change

Sara and Tanvi are taking part in an orienteering race.

The start and finish points of the race are the same. They will take different routes from the start point to get to the same last marker point. They then head directly back to the start/finish point.

Here is a diagram showing some information about Sara’s route.

Show that the distance Sara travels from the last marker back to the finish point is

7∙7 km, correct to 1 decimal place.

17 17

The diagram below shows some information about Tanvi’s route.

Tanvi started her race by running 6∙3 km on a bearing of 120º to point A. She then ran 2∙1 km to reach the last marker.

Calculate the bearing of the last marker from point A.

The map that Sara and Tanvi were each given before the start of the race was 33 cm long. Tanvi decided to reduce the size of her map to a mathematically similar size.

The area of Tanvi’s map is now 19% less than the area of Sara’s map.

Calculate the length of Tanvi’s map.

Sara’s distance from the start, for the first part of her race, is shown in the graph below.

Estimate Sara’s speed 20 minutes after the start of the race.

Give your answer in miles per hour.

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.

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).

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.

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.

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.

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.