Unit 9
8 marks · EasyA solid cuboid has sides 15 cm, 8 cm, and 6 cm. Its dimensions are measured to the nearest cm. A shop packs chocolates of two sizes. The company checks the distributions of their box heights with histograms, shown below.
Using the cuboid shown, explain how you would find the length of the main body diagonal from one corner to the farthest corner. State the formula in terms of the side lengths.
Calculate the length of the body diagonal of the cuboid. Give your answer to 3 significant figures.
Interpret the histogram above. Compare the number and spread of shorter versus taller boxes. What does the distribution tell you about the two products?
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) Use 3D Pythagoras: diagonal = , where a, b, c are the side lengths.
(b) 18.4 cm
(c) Most boxes have heights between 6 and 9 cm (highest frequency density is 4), so the distribution is concentrated at shorter heights. Fewer boxes are in the largest class (15 < h ≤ 18 cm, frequency density 1), showing taller boxes are less common. This suggests the company mainly makes smaller boxes with a few larger ones.
Common mistakes: Leaving out one dimension in diagonal formula; Calculating only the face diagonal, not the main body diagonal; Comparing only frequencies, not frequency density or spread
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · States correct formula for 3D body diagonal:
B1 (a) · 1 mark · Describes using all three side lengths (no omission of any dimension)
M2 (b) · 1 mark · Substitutes values:
A1 (b) · 1 mark · Accurately calculates and rounds answer to 3 s.f.: 18.0 cm
M3 (c) · 1 mark · Identifies the class with highest frequency density (6 < h ≤ 9 cm)
B2 (c) · 2 marks · Compares spread and frequency density of tall vs short boxes
B3 (c) · 1 mark · Concludes that most boxes are short, and taller boxes are less common based on the histogram
Worked solution
(a) To find the main diagonal of a cuboid, apply Pythagoras’ theorem in three dimensions: , where a, b, c are the edge lengths.
(b) cm (to 3 s.f.)
(c) The tallest bar represents the class 6 < h ≤ 9 cm, frequency density 4. This means most boxes are in the smallest height group. The class 9 < h ≤ 12 cm has a lower frequency density (2), and 12 < h ≤ 15 cm is in-between (3). The largest height range, 15 < h ≤ 18 cm, has the smallest frequency density (1), showing tall boxes are uncommon.
Therefore, the distribution is skewed towards shorter boxes; the company mostly produces smaller boxes, with only a few tall (larger) ones.