Unit 8
10 marks · HardThis question asks you to convert units of area and calculate the magnitude of a vector.
A farmer's rectangular field measures 150 m by 80 m. Express the area in square kilometres. Give your answer in standard form.
A cow walks from one corner of the field at (0,0) to the corner at (150,80). Write the displacement vector as a column vector (in metres).
Calculate the magnitude of this displacement, giving your answer to 2 decimal places. State the unit.
The same cow walks to the midpoint of the opposite side, at position (75, 80). Find the magnitude of the displacement from the start to this midpoint. Give your answer as an exact surd.
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) 1.2 × 10^{-2} km²
(b) m
(c) 170.88 m
(d) m
Common mistakes: Multiplying by 100 for area rather than dividing by 1,000,000 for km².; Forgetting surd form/decimal places requirements.
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Converts from m² to km² by dividing by 1,000,000.
A1 (a) · 1 mark · Area in correct standard form.
A2 (a) · 1 mark · Final answer written as 1.2 × 10^{-2} km².
M2 (b) · 2 marks · Identifies correct column vector for displacement from (0,0) to (150,80).
M3 (c) · 1 mark · Applies Pythagoras’ theorem to components. (follow-through)
A3 (c) · 1 mark · Correct value to 2 decimal places, unit stated. (follow-through)
M4 (d) · 2 marks · Identifies correct vector as (75,80), applies Pythagoras.
A4 (d) · 1 mark · Gives answer as exact surd form. (follow-through)
Worked solution
a) Area = 150 m × 80 m = 12,000 m². 1 km² = 1,000,000 m². So 12,000 ÷ 1,000,000 = 0.012 km² = 1.2 × 10^{-2} km².
b) Displacement vector = m.
c) Magnitude is 170.88 m (rounded to 2 d.p.).
d) Vector to (75,80) is , so magnitude is m.