Exam questions · Maths · Graphs
Real-Life Graphs
- 6 exam questions
- 22 marks
- 9 quick checks
-
1 Work out [6 marks]
The graph shows the velocity of a cyclist during a 14 second journey. (a) Work out the acceleration of the cyclist in the first 4 seconds. (2 marks) (b) Work out the distance travelled in the first 4 seconds. (2 marks) (c) Work out the total distance travelled in the 14 seconds. (2 marks)
Show answerHide answer
Model answer
(a) Acceleration \(= \dfrac{12}{4} = 3\) m/s\(^2\). (b) The distance is the area of the triangle, \(\dfrac{1}{2} \times 4 \times 12 = 24\) m. (c) The shape is a trapezium with parallel sides 14 and 6 and height 12, so the area is \(\dfrac{1}{2}(14 + 6) \times 12 = 120\) m.
Mark scheme
- (a) \(\dfrac{12}{4}\) — M1
- (a) 3 m/s\(^2\) — A1
- (b) \(\dfrac{1}{2} \times 4 \times 12\) — M1
- (b) 24 m — A1
- (c) \(\dfrac{1}{2}(14 + 6) \times 12\) or the areas of the three parts added — M1
- (c) 120 m — A1
-
2 Work out [3 marks]
A taxi company charges \(\pounds 4\) plus \(\pounds 3\) for each kilometre. (a) Work out the cost of a journey of 6 km. (1 mark) (b) Write a formula for the cost, \(C\) pounds, of a journey of \(d\) kilometres. (2 marks)
Show answerHide answer
Model answer
(a) \(4 + 3 \times 6 = \pounds 22\). (b) \(C = 3d + 4\).
Mark scheme
- (a) \(\pounds 22\) — B1
- (b) \(3d\) seen or \(C = \ldots + 4\) — M1
- (b) \(C = 3d + 4\) — A1
-
3 Work out [3 marks]
A water tank contains 20 litres of water. Water is added at a steady rate of 5 litres each minute. The graph of the volume \(V\) litres against time \(t\) minutes is a straight line. (a) Work out the volume of water in the tank after 12 minutes. (2 marks) (b) What does the gradient of the graph represent? (1 mark)
Show answerHide answer
Model answer
(a) \(V = 5t + 20\), so after 12 minutes \(V = 5 \times 12 + 20 = 80\) litres. (b) The gradient, 5, is the rate at which water is added, in litres per minute.
Mark scheme
- (a) \(5 \times 12 + 20\) — M1
- (a) 80 litres — A1
- (b) The rate of filling, 5 litres per minute — C1
-
4 Work out [4 marks]
A train starts from rest. It speeds up at a steady rate until it reaches 30 m/s after 15 seconds. It then slows down at a steady rate and stops 10 seconds later. (a) Work out the acceleration of the train in the first 15 seconds. (1 mark) (b) Work out the total distance travelled by the train. (3 marks)
Show answerHide answer
Model answer
(a) \(\dfrac{30}{15} = 2\) m/s\(^2\). (b) The velocity-time graph is a triangle with base \(15 + 10 = 25\) seconds and height 30 m/s, so the distance is \(\dfrac{1}{2} \times 25 \times 30 = 375\) m.
Mark scheme
- (a) 2 m/s\(^2\) — B1
- (b) Base 25 seen, or the areas of two triangles — M1
- (b) \(\dfrac{1}{2} \times 25 \times 30\) — M1
- (b) 375 m — A1
-
5 Explain [2 marks]
Water is poured at a steady rate into a vase. The vase is narrow at the bottom and gets wider towards the top. Describe how the graph of the depth of water against time changes as the vase fills.
Show answerHide answer
Model answer
At first the water is in a narrow part, so the depth rises quickly and the graph is steep. As the vase gets wider, the depth rises more slowly, so the graph gets less steep and flattens.
Mark scheme
- The graph starts steep — C1
- and gets less steep, or flatter, as the vase fills — C1
-
6 Work out [4 marks]
A car moves so that its distance \(s\) metres from a point after \(t\) seconds is given by \(s = 5t^2\). (a) Complete the table for \(t = 0, 1, 2, 3, 4\). (2 marks) (b) By finding the distance travelled between \(t = 1\) and \(t = 3\), estimate the speed of the car at \(t = 2\). (2 marks)
Show answerHide answer
Model answer
(a) The values are \(0, 5, 20, 45, 80\). (b) Between \(t = 1\) and \(t = 3\) the car travels \(45 - 5 = 40\) m in 2 seconds, so the speed is about \(\dfrac{40}{2} = 20\) m/s.
Mark scheme
- (a) At least three correct values — M1
- (a) \(0, 5, 20, 45, 80\) — A1
- (b) \(\dfrac{45 - 5}{3 - 1}\) — M1
- (b) 20 m/s — A1
Quick check
-
1
On a conversion graph, 5 miles is about 8 km. About how many kilometres is 30 miles?
Show answerHide answer
B: 48 km
30 miles is 6 lots of 5 miles, so \(6 \times 8 = 48\) km.
-
2
What does the gradient of a velocity-time graph show?
Show answerHide answer
A: Acceleration
Gradient is change in velocity divided by time, which is acceleration.
-
3
What does the area under a velocity-time graph show?
Show answerHide answer
D: Distance travelled
Velocity multiplied by time gives distance.
-
4
A car speeds up from 0 to 12 m/s in 4 seconds. What is its acceleration?
Show answerHide answer
C: 3 m/s\(^2\)
\(\dfrac{12}{4} = 3\).
-
5
A taxi costs \(\pounds 3\) plus \(\pounds 2\) for each kilometre. What is the cost of a 7 km journey?
Show answerHide answer
B: \(\pounds 17\)
\(3 + 2 \times 7 = 17\).
-
6
On a graph of taxi cost against distance, what does the \(y\)-intercept mean?
Show answerHide answer
A: The fixed starting charge
The intercept is the cost for 0 km.
-
7
A container gets wider towards the top and is filled at a steady rate. What happens to the depth-time graph?
Show answerHide answer
D: It rises more and more slowly, so it flattens
The wider the container, the more slowly the depth rises.
-
8
A velocity-time graph is a triangle that rises from 0 to 8 m/s in 5 seconds. What distance does it show?
Show answerHide answer
C: 20 m
Area \(= \dfrac{1}{2} \times 5 \times 8 = 20\).
-
9
How can you estimate the speed at one moment from a curved distance-time graph?
Show answerHide answer
B: Draw a tangent and find its gradient
The gradient of the tangent is the rate of change at that point.