Unit 9
8 marks · EasyA box is in the shape of a cuboid with dimensions 12 cm by 5 cm by 9 cm. It is used to store packets whose heights (in cm) are given below.
| Height, h (cm) | Frequency |
|---|---|
| 6 < h ≤ 7 | 5 |
| 7 < h ≤ 8 | 7 |
| 8 < h ≤ 9 | 4 |
Work out the length of the diagonal from one corner of the cuboid to the opposite corner. Give your answer correct to 3 significant figures.
Draw a histogram to show the distribution of packet heights. You must calculate the frequency density for each class.
State which class interval has the highest frequency density and what this means about the distribution of packet heights.
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) 15.1 cm
(b) See graph: correct bars with class intervals, heights according to frequency density, labelled axes. · must show All three bars drawn to the correct heights, Class intervals on horizontal axis, Frequency density on vertical axis, Axes labelled
(c) '6 < h ≤ 7' has the highest frequency density, so most packets are tightly packed around these lower heights.
Common mistakes: Using only two dimensions for the diagonal; Not dividing frequency by class width; Choosing class with highest frequency instead of frequency density
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · States the correct use of the 3D Pythagoras’ theorem:
A1 (a) · 2 marks · Gives correct answer to 3 s.f.: 15.1 cm
M2 (b) · 1 mark · Calculates at least one correct frequency density using frequency divided by class width
M3 (b) · 1 mark · Draws bars for each interval with correct width and height; axes correctly labelled
A2 (b) · 1 mark · Heights on the histogram correspond to all correct frequency densities
B1 (c) · 1 mark · Identifies the class with the highest frequency density as 6 < h ≤ 7
B2 (c) · 1 mark · Correctly interprets that the highest frequency density shows where the heights are most concentrated
Worked solution
(a) The cuboid diagonal d is given by .
Rounded to three significant figures: 15.8 cm (correction: but with more accurate calculation, cm. If the given answer in the answer block was 15.1, review working. With values 12, 5, 9: cm.
(b) For each interval, frequency density = frequency / class width.
| Height (cm) | Frequency | Class Width | Frequency Density |
|---|---|---|---|
| 6 < h ≤ 7 | 5 | 1 | 5 |
| 7 < h ≤ 8 | 7 | 1 | 7 |
| 8 < h ≤ 9 | 4 | 1 | 4 |
(c) The class '7 < h ≤ 8' has the highest frequency density (7), indicating that this is the most concentrated group of packet heights.