Unit 9
11 marks · HardA curved graph is shown below. The graph shows the distance, metres, travelled by a cyclist after seconds.
Use the graph to estimate the gradient of the curve at seconds. Give your answer in .
A group of students all cycle the same route. The frequency table below shows the time each student takes to finish the route.
| Time, t (seconds) | Frequency |
|---|---|
| 0 < t \leq 4 | 4 |
| 4 < t \leq 8 | 7 |
| 8 < t \leq 10 | 9 |
| 10 < t \leq 16 | 10 |
Draw a histogram to represent the data in the table. Your histogram should have frequency density on the vertical axis.
By referring to your histogram and the graph in part (a), compare how the cyclist's time at seconds relates to the overall distribution of times for the group. Write two facts comparing these results.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) 8 m/s · must show Correct use of a tangent at t=6, Working with rise/run (change in s/change in t), Accurate calculation of gradient in m/s, Appropriate units given
(b) Histogram with correct bars for each class: heights 1, 1.75, 4.5, 1.67, corresponding to the four classes, frequency density on y-axis, t (s) on x-axis, correct labels, no gaps between bars. · must show Bars heights: 1, 1.75, 4.5, 1.67, Accurate class boundaries, Frequency density and axis labels, No gaps between bars
(c) The cyclist had reached 18 m at 6 s, which suggests a faster-than-average time compared to most students. Only a small number of students completed the route in less than 6 or 8 seconds, as seen in the histogram. · must show Clear comparison to group distribution, Reference to histogram shape, Reference to curve result at t=6
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Draw or use a correct tangent at on the curve.
M2 (a) · 1 mark · Clearly identify and use two points on the tangent to calculate change in and change in .
A1 (a) · 1 mark · Correct value for gradient from change in . (Accept 6.5 to 9.5 m/s for estimation.) (follow-through)
A2 (a) · 1 mark · Gives correct units (m/s) for the gradient.
M3 (b) · 1 mark · Calculate correct class widths for each group in the table.
M4 (b) · 1 mark · Calculate frequency density for each class (accept follow-through for errors in widths). (follow-through)
A3 (b) · 1 mark · Draws histogram with correct bar heights for frequency densities corresponding to each class width. (follow-through)
A4 (b) · 1 mark · Axes fully labelled with correct scale, no gaps between bars, correct class boundaries.
B1 (c) · 1 mark · States a correct fact comparing the cyclist's time/curve to the histogram distribution.
B2 (c) · 1 mark · States a second correct comparison or interpretation linking the curve and histogram.
B3 (c) · 1 mark · Both facts are clear and relevant to distribution and result at s.
Worked solution
(a) To estimate the gradient of the curve at s:
- Draw a tangent to the curve at . Suppose the tangent meets the curve near and .
Alternatively, if the tangent meets at points and :
Depending on the drawn tangent, a reasonable estimate is about 4 to 8 m/s. For optimal method, gradient ≈ 8 m/s is accepted.
(b) Find class widths and frequency densities:
| Class | Frequency | Width | Freq. Density |
|---|---|---|---|
| 0 < t \leq 4 | 4 | 4 | 1 |
| 4 < t \leq 8 | 7 | 4 | 1.75 |
| 8 < t \leq 10 | 9 | 2 | 4.5 |
| 10 < t \leq 16 | 10 | 6 | 1.67 |
Draw histogram: x-axis: t (s) from 0 to 16, bars covering intervals with heights 1, 1.75, 4.5, 1.67. y-axis: Frequency density.
(c) Comparison:
- At s, the cyclist is at 18 m (from the curve), which is the time to cover most of the route quickly.
- The histogram shows most students took more than 6 s; only those in the first two groups completed the route within 8 s.
Therefore, the cyclist was quicker than most students; only a small proportion finished in the fastest time as shown by the histogram's smaller bars for the shortest times.