NBT Mathematical Modelling 2 — Questions and Answers
Question 1: A farmer has 120 metres of fencing. He wants to enclose a rectangular area against a wall (needing only 3 sides of fencing). What dimensions maximise the area?
- 30 m × 60 m (Correct answer)
- 40 m × 40 m
- 60 m × 30 m
- Both A and C
Correct answer: 30 m × 60 m
Let width = x. Then length = 120 - 2x. Area A = x(120 - 2x) = -2x² + 120x. Maximum at x = -120/(2(-2)) = 30. So width = 30 m and length = 60 m.
Question 2: The temperature in Celsius can be converted to Fahrenheit using F = 1.8C + 32. At what temperature are the Celsius and Fahrenheit readings equal?
- -40° (Correct answer)
- 0°
- 32°
- 100°
Correct answer: -40°
Set F = C: C = 1.8C + 32 → -0.8C = 32 → C = -40. At -40°, both scales read the same. This is the unique intersection point of the conversion function with y = x.
Question 3: A company's profit P (in thousands of rands) is modelled by P = -2x² + 24x - 40, where x is the number of units sold (in thousands). How many units must be sold to break even?
- 2 000 or 10 000 units (Correct answer)
- 4 000 or 8 000 units
- 6 000 units only
- 12 000 units
Correct answer: 2 000 or 10 000 units
Break even when P = 0: -2x² + 24x - 40 = 0 → x² - 12x + 20 = 0 → (x - 2)(x - 10) = 0. So x = 2 or x = 10 (thousands), meaning 2 000 or 10 000 units.
Question 4: A swimming pool is being filled. After t minutes, the volume V (litres) is V = 50t. How long will it take to fill a 5 000-litre pool?
- 10 minutes
- 50 minutes
- 100 minutes (Correct answer)
- 250 minutes
Correct answer: 100 minutes
Set V = 5 000: 5 000 = 50t → t = 100 minutes. At a constant rate of 50 litres per minute, 5 000 litres takes 100 minutes.
Question 5: The number of bacteria in a culture doubles every 3 hours. Starting with 500 bacteria, how many will there be after 12 hours?
- 2 000
- 4 000
- 6 000
- 8 000 (Correct answer)
Correct answer: 8 000
After 12 hours, there are 12/3 = 4 doublings. Population = 500 × 2⁴ = 500 × 16 = 8 000 bacteria.
Question 6: An investment of R10 000 earns compound interest at 8% per annum compounded quarterly. What is the value after 2 years?
- R11 600
- R11 716.59 (Correct answer)
- R11 664.64
- R11 717.39
Correct answer: R11 716.59
Quarterly compounding: A = 10 000(1 + 0.08/4)^(4×2) = 10 000(1.02)^8 = 10 000 × 1.171659 ≈ R11 716.59.
NBT Mathematical Modelling 2
A farmer has 120 metres of fencing.
He wants to enclose a rectangular area against a wall (needing only 3 sides of fencing).
What dimensions maximise the area?