Year 11 rates of change and areas under graphs worksheet with answers to download as PDF

This Year 11 rates of change and areas under graphs worksheet with answers has 20 questions for pupils aged 15-16, from short skills practice to real-life problems. The worked answers are on a separate page, so you can try every question first.

Question 1 – Reading a distance-time graph (Section A · Skills practice)
Sam cycles to a village and back. The points on his distance-time graph are joined by straight lines.
Time (hours) 0 2 2.5 4.5
Distance from home (km) 0 60 60 0
1. Work out his speed for the first part of the journey. [2 marks]
2. What happened between 2 and 2.5 hours? [1 mark]
3. Work out his speed on the way home. [2 marks]
4. Work out his average speed for the whole journey. [3 marks]
Question 2 – Units of speed and acceleration (Section A · Skills practice)
Convert and calculate.
1. Change 72 km/h into m/s. [2 marks]
2. Change 15 m/s into km/h. [2 marks]
3. A car increases its speed from 0 to 24 m/s in 8 seconds. Work out its acceleration. [2 marks]
4. State the units of acceleration. [1 mark]
Question 3 – Area under a velocity-time graph (Section A · Skills practice)
A train travels between two stations. Its velocity-time graph has these corner points: \((0,0)\), \((10,20)\), \((30,20)\), \((40,0)\), with time in seconds and velocity in m/s.
1. Work out the acceleration for the first 10 seconds. [2 marks]
2. Work out the distance travelled while the speed is constant. [2 marks]
3. Work out the total distance between the stations. [4 marks]
4. Work out the average speed for the journey. [2 marks]
Question 4 – Gradient as a rate (Section A · Skills practice)
Water flows into a tank at a steady rate. After 5 minutes there are 150 litres in the tank. The graph of volume against time is a straight line through the origin.
1. Work out the gradient of the graph and state its units. [2 marks]
2. What does the gradient represent? [1 mark]
3. How long does it take to fill a 450-litre tank? [2 marks]
4. The flow is reduced to 20 litres per minute. How does the graph change? [1 mark]
Question 5 – Rates in different units (Section A · Skills practice)
A tap delivers water at 12 litres per minute.
1. Write the rate in litres per second. [2 marks]
2. Write the rate in litres per hour. [1 mark]
3. How long does the tap take to fill a 30-litre bucket? [1 mark]
4. A hose delivers 0.5 litres per second. Which is faster? [2 marks]
Question 6 – Estimating distance with trapezia (Higher) (Section A · Skills practice)
The speed of a car is recorded every 2 seconds.
Time (s) 0 2 4 6
Speed (m/s) 0 6 10 12
1. Use three trapezia to estimate the distance travelled in 6 seconds. [4 marks]
2. Is the estimate an overestimate or an underestimate if the graph curves like a "bend" (concave down)? Explain. [2 marks]
3. Work out the average speed over the 6 seconds. [1 mark]
Question 7 – A run to the park (Section B · Real-life practice)
Zoe runs from home to the park, rests, then runs on to the shops. Her distance-time graph has corner points \((0,0)\), \((1.5,45)\), \((2,45)\), \((3.5,90)\) with time in hours and distance in km.
1. Work out her speed in each running section. [3 marks]
2. For how long does she rest? [1 mark]
3. Work out her average speed for the whole trip including the rest. [3 marks]
4. Work out her average speed while actually running. [3 marks]
Question 8 – A bus journey (Section B · Real-life practice)
A bus leaves a station and travels along a route. Corner points of its distance-time graph: \((0,0)\), \((0.5,15)\), \((1,15)\), \((2,60)\) with time in hours and distance in km.
1. Work out the speed for each moving section. [3 marks]
2. What does the horizontal section show? [1 mark]
3. Work out the average speed for the whole journey. [2 marks]
4. The bus timetable says the journey should take 1.5 hours. How much late is the bus? [2 marks]
Question 9 – Car acceleration (Section B · Real-life practice)
A sports car accelerates from rest to 100 km/h in 8 seconds.
1. Change 100 km/h into m/s to 2 decimal places. [2 marks]
2. Work out the average acceleration. [2 marks]
3. The car continues to accelerate at the same rate. How fast is it going after 12 seconds? Give your answer in km/h. [3 marks]
4. Work out the distance covered in the first 8 seconds, assuming constant acceleration. [3 marks]
Question 10 – A lift in a tall building (Section B · Real-life practice)
A lift accelerates uniformly at 2 m/s² for 3 seconds, then travels at constant speed for 5 seconds and then slows uniformly to rest in 4 seconds.
1. Find the top speed. [1 mark]
2. Sketch the velocity-time graph (describe its corner points). [3 marks]
3. Work out the total distance. [4 marks]
4. Work out the deceleration in the last section. [2 marks]
Question 11 – Filling a swimming pool (Section B · Real-life practice)
A pool is 6 m long, 3 m wide and 1.2 m deep. A pump delivers 25 litres per minute. 1 m³ \(=1\,000\) litres.
1. Work out the volume of the pool in litres. [3 marks]
2. How long does it take to fill the pool? Give your answer in hours and minutes. [3 marks]
3. A second pump delivers 50 litres per minute. How long do the two pumps take together? [3 marks]
Question 12 – Cycling speeds (Section B · Real-life practice)
A cyclist travels at 24 km/h.
1. Write the speed in m/s to 2 decimal places. [2 marks]
2. How long does she take to travel 500 m? [2 marks]
3. How far does she travel in 45 minutes? [2 marks]
4. A runner travels 5 m/s. Who is faster and by how much in km/h? [3 marks]
Question 13 – Population change (Section B · Real-life practice)
The population of a town was 50 000 in 2010 and 56 000 in 2020. Assume the graph of population against year is a straight line.
1. Work out the gradient and state its meaning. [3 marks]
2. Predict the population in 2030. [2 marks]
3. The mayor says the town grows by 1 000 people per year. Is this correct? [1 mark]
4. In which year does the population reach 60 000? [2 marks]
Question 14 – Cooling coffee (Section B · Real-life practice)
A cup of coffee at 80°C cools to 50°C after 6 minutes.
1. Work out the average rate of change of temperature. [2 marks]
2. Explain the sign of your answer. [1 mark]
3. If it cooled at this steady rate, when would it reach 20°C? [2 marks]
4. In reality the cooling slows down. What would the graph look like? [2 marks]
Question 15 – A three-stage journey (Section C · Problem solving)
A car accelerates from rest to 12 m/s in 4 seconds, travels at 12 m/s for 6 seconds and then slows to rest in 3 seconds.
1. Work out the acceleration in the first stage. [2 marks]
2. Work out the deceleration in the last stage. [2 marks]
3. Work out the total distance. [4 marks]
4. Work out the average speed for the whole journey. [2 marks]
Question 16 – Overtaking (Section C · Problem solving)
Car A leaves a town at 50 km/h. Car B leaves the same town half an hour later at 75 km/h on the same road.
1. Write the distance of each car from the town, \(s\) km, after \(t\) hours since A left. [3 marks]
2. Find when B catches up with A. [3 marks]
3. How far from the town is this? [1 mark]
4. Describe how the two lines would look on a distance-time graph. [2 marks]
Question 17 – A curved velocity-time graph (Higher) (Section C · Problem solving)
The velocity of a cyclist is recorded every 2 seconds.
Time (s) 0 2 4 6 8
Velocity (m/s) 0 5 8 10 11
1. A tangent to the curve at \(t=4\) passes through \((2,3)\) and \((6,11)\). Find the acceleration at \(t=4\). [3 marks]
2. Use four trapezia to estimate the distance travelled. [4 marks]
3. Is it an overestimate or underestimate? Explain. [2 marks]
Question 18 – A leaking tank (Section C · Problem solving)
A tap fills a 400-litre tank at 15 litres per minute but a leak loses 3 litres per minute.
1. Work out the net rate of filling. [1 mark]
2. How long does it take to fill the tank? [2 marks]
3. How many litres are lost to the leak in this time? [2 marks]
4. The leak is mended after 10 minutes. How long does it take to fill from that point? [3 marks]
Question 19 – A cheetah (Section C · Problem solving)
A cheetah runs 100 m in 4.2 seconds.
1. Work out its average speed in m/s. [2 marks]
2. Convert the answer to km/h. [2 marks]
3. A car travels at 85 km/h. Which is faster? [1 mark]
4. The cheetah can keep this speed for 20 seconds. How far does it go? [2 marks]
Question 20 – Challenge: how good is the estimate? (Section D · Challenge)
The velocity of a particle is \(v=t(8-t)\) m/s for \(0\le t\le8\). The true distance travelled is \(\dfrac{256}3\) m.
1. Complete a table for \(t=0,2,4,6,8\). [2 marks]
2. Use four trapezia to estimate the distance. [4 marks]
3. Find the percentage error in your estimate. [3 marks]
4. How could the estimate be improved? [1 mark]

Time (hours)	0	2	2.5	4.5
Distance from home (km)	0	60	60	0

Time (s)	0	2	4	6
Speed (m/s)	0	6	10	12

Time (s)	0	2	4	6	8
Velocity (m/s)	0	5	8	10	11

How to use this Year 11 rates of change and areas under graphs worksheet with answers

First, read the rates of change and areas under graphs lesson so that the methods are fresh in your mind. Then answer the questions in order, because they start easy and get harder. Show your working on paper; afterwards, compare it with the worksheet answers. If a question goes wrong, look back at the matching example before you try again. Finally, when you feel confident, try the rates of change and areas under graphs test or the quiz.

You can also download this Year 11 rates of change and areas under graphs worksheet with answers as a PDF and print it for homework. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 11 rates of change and areas under graphs worksheet with answers prepares you well for the test.

Rates of change and areas under graphs – Year 11 GCSE maths worksheet with answers to download as PDF

Question 1 – Reading a distance-time graph (Section A · Skills practice)

Question 2 – Units of speed and acceleration (Section A · Skills practice)

Question 3 – Area under a velocity-time graph (Section A · Skills practice)

Question 4 – Gradient as a rate (Section A · Skills practice)

Question 5 – Rates in different units (Section A · Skills practice)

Question 6 – Estimating distance with trapezia (Higher) (Section A · Skills practice)

Question 7 – A run to the park (Section B · Real-life practice)

Question 8 – A bus journey (Section B · Real-life practice)

Question 9 – Car acceleration (Section B · Real-life practice)

Question 10 – A lift in a tall building (Section B · Real-life practice)

Question 11 – Filling a swimming pool (Section B · Real-life practice)

Question 12 – Cycling speeds (Section B · Real-life practice)

Question 13 – Population change (Section B · Real-life practice)

Question 14 – Cooling coffee (Section B · Real-life practice)

Question 15 – A three-stage journey (Section C · Problem solving)

Question 16 – Overtaking (Section C · Problem solving)

Question 17 – A curved velocity-time graph (Higher) (Section C · Problem solving)

Question 18 – A leaking tank (Section C · Problem solving)

Question 19 – A cheetah (Section C · Problem solving)

Question 20 – Challenge: how good is the estimate? (Section D · Challenge)

How to use this Year 11 rates of change and areas under graphs worksheet with answers

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