(a) Jianyu records the time, in seconds, that some cars take to travel 195m. The box and whisker plot shows this information.
(i) Find the median time.
(ii) Find the interquartile range.
(iii) Find the difference between the average speed of the fastest car and the average speed of the slowest car. Give your answer in kilometres per hour.
(b) Matilda records the distances that 80 different cars can travel with a full tank of fuel. The table shows this information.
(i) Write down the class interval that contains the median.
(ii) Calculate an estimate of the mean.
(iii) A histogram is drawn to show the information in the table. The height of the bar for the interval \(250 \leqslant d \leqslant 300\) is 2.8 cm. Calculate the height of the bar for each of the following intervals.
(iv) Two of the 80 cars are chosen at random. Find the probability that, with a full tank of fuel, one of the cars can travel more than 450km and the other car can travel not more than 300 km.
▶️ Answer/Explanation
(a)(i) Ans: 9.3
The median is the middle line in the box plot, which is at 9.3 seconds.
(a)(ii) Ans: 3.4
Interquartile range (IQR) = Upper Quartile (11.6) – Lower Quartile (8.2) = 3.4 seconds.
(a)(iii) Ans: 63
Fastest car speed = 195/6 = 32.5 m/s. Slowest car speed = 195/13 = 15 m/s. Difference = 17.5 m/s = 63 km/h.
(b)(i) Ans: 420 < d < 450
Median is at the 40th value, which falls in the 420-450 km class interval.
(b)(ii) Ans: 411.25
Mean = (Sum of midpoints × frequencies) / total frequency = 32900/80 = 411.25 km.
(b)(iii) Ans: 2.6, 19, 14
Using the scaling factor 20, heights are calculated as (frequency/class width) × 20.
(b)(iv) Ans: 7/158
Probability = (20/80 × 7/79) + (7/80 × 20/79) = 280/6320 = 7/158.
Marco is making patterns with grey and white circular mats.
The patterns form a sequence.
Marco makes a table to show some information about the patterns.
(a) Complete the table for Pattern 5.
(b) Find an expression, in terms of n, for the number of grey mats in Pattern n.
(c) Marco makes a pattern with 24 grey mats. Find the total number of mats in this pattern.
(d) Marco needs a total of 6 mats to make the first pattern. He needs a total of 16 mats to make the first two patterns. He needs a total of \(\frac{1}{6}n^{3}+an^{2}+bn\) mats to make the first n patterns. Find the value of a and the value of b.
▶️ Answer/Explanation
(a) Ans: 18, 28
For Pattern 5, grey mats follow the sequence \(3n + 3\) → \(3(5) + 3 = 18\). Total mats follow \(n^2 + 4n\) → \(5^2 + 4(5) = 25 + 20 = 45\). White mats = Total – Grey = \(45 – 18 = 27\).
(b) Ans: \(3n + 3\)
Observing the pattern: 6, 9, 12, 15, … grey mats increase by 3 each time. Thus, the expression is linear: \(3n + 3\).
(c) Ans: 45
Set \(3n + 3 = 24\) → \(n = 7\). Total mats for Pattern 7: \(7^2 + 4(7) = 49 + 28 = 77\). White mats = \(77 – 24 = 53\).
(d) Ans: \(a = \frac{3}{2}\), \(b = \frac{13}{3}\)
Using given totals: For \(n = 1\), \(\frac{1}{6} + a + b = 6\). For \(n = 2\), \(\frac{8}{6} + 4a + 2b = 16\). Solve the system to find \(a\) and \(b\).