Unit 9
8 marks · EasyA regular square-based pyramid stands on ABCD, where each side of the base is 10 cm. The vertex, V, is 14 cm above the centre of the base. The masses (in g) of stones used to make several pyramids are grouped into classes as shown.
| Mass, m (g) | Frequency |
|---|---|
| 200 < m ≤ 300 | 4 |
| 300 < m ≤ 500 | 9 |
| 500 < m ≤ 800 | 7 |
Calculate the slant height of the pyramid (from the centre of one base edge to the vertex). Give your answer to 3 significant figures.
Calculate the frequency density for each class in the mass table.
Estimate the proportion of pyramids with mass greater than 500 g.
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) 17.1 cm
(b) See table: 0.04, 0.045, 0.0233 · must show Correct frequency density for each class
(c) 7/20 = 0.35 or 35% · must show Proportion as decimal or percent
Common mistakes: Finding the height along the perpendicular instead of slant height; Not dividing by correct class width; Omitting to sum frequencies for 'greater than' class
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Correct sketch or identification of triangle for slant height: half-base side and pyramid height
M2 (a) · 1 mark · Uses Pythagoras:
A1 (a) · 1 mark · Final answer rounded to 3 s.f. (17.1 cm)
M3 (b) · 1 mark · Calculates at least one correct frequency density: frequency/class width
A2 (b) · 1 mark · All three correct: 0.04, 0.045, 0.0233
M4 (c) · 1 mark · Adds total for 500 < m ≤ 800 class: 7
M5 (c) · 1 mark · Finds total number of pyramids (4+9+7=20)
A3 (c) · 1 mark · Proportion as correct fraction or percentage (0.35 or 35%)
Worked solution
(a) The distance from the centre of the base to the centre of an edge is 5 cm (half a side). The slant height, s, is:
But from the centre to a vertex (if asked for vertex, not edge), it would be cm. Here, as the question says "to the centre of one base edge," the intended answer is cm.
Rounded to 3 s.f.: 14.9 cm.
(b) Class widths and frequency densities:
| Mass, m (g) | Frequency | Class Width | Frequency Density |
|---|---|---|---|
| 200 < m ≤ 300 | 4 | 100 | 0.04 |
| 300 < m ≤ 500 | 9 | 200 | 0.045 |
| 500 < m ≤ 800 | 7 | 300 | 0.0233 |
(c) Number of pyramids with mass > 500 g: 7
Total number: 4+9+7=20