Unit 8

11 marks · Hard

This question is about solving problems involving unit conversions and vectors.

(a)

The rectangular floor of a hall measures 48 m in length and 32 m in width. Convert the area of the floor into square centimetres. Give your answer in standard form.

[3 marks]

(b)

A cleaner moves from point A (at one corner of the hall) to B (at the diagonally opposite corner). Express the displacement vector from A to B as a column vector in metres.

[2 marks]

(c)

Calculate the magnitude of her displacement from part (b). Give your answer to 3 significant figures. State the unit.

[2 marks]

(d)

Suppose another cleaner moves from A to C, where C is halfway along the width from A (i.e., 16 m from A along the width), and then to D, which is halfway along the length from C (i.e., 24 m farther in the length direction). Express the total displacement vector from A to D as a column vector in metres.

[2 marks]

(e)

Calculate the magnitude of the displacement from A to D. Give your answer as a surd or to 2 decimal places.

[2 marks]

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) 1.54 × 10^8 cm²

(b) m

(d) m

(e) m or 28.8 m

Common mistakes: Multiplying by 100 instead of 10000 for area conversion; Picking the wrong difference for vector components; Forgetting to use Pythagoras to find magnitude

Show mark scheme and worked solution

Marking points

M1 (a) · 1 mark · Correctly converts metres to cm and squares values to convert m² to cm².

A1 (a) · 1 mark · Arrives at 1,536,000,000 cm² or correct standard form for area.

A2 (a) · 1 mark · Expresses answer in standard form.

M2 (b) · 2 marks · States displacement vector from (0,0) to (48,32) as column vector in metres.

M3 (c) · 1 mark · Uses Pythagoras’ theorem to find magnitude: sqrt(48^2 + 32^2).

A3 (c) · 1 mark · Final answer correct to 3 significant figures with correct units. (follow-through)

M4 (d) · 2 marks · Finds correct column vector for displacement from A to D: (24,16).

M5 (e) · 1 mark · Correct application of Pythagoras for this displacement vector.

A4 (e) · 1 mark · Gives answer in surd form or to 2 decimal places, with unit. (follow-through)

Worked solution

a) Area = 48 m × 32 m = 1536 m². 1 m² = 10,000 cm², so 1536 m² = 1536 × 10,000 = 15,360,000 cm² = 1.54 × 10⁸ cm² (to 3 s.f.).

b) Vector from (0,0) at A to (48,32) at B is m.

c) Magnitude from part b: m (to 3 s.f.).

d) D is 24 m along length and 16 m along width, so vector is m.

e) Magnitude = m (to 2 d.p.).