NBT Mathematical Modelling Flashcards

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?

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.

The temperature in Celsius can be converted to Fahrenheit using F = 1.8C + 32. At what temperature are the Celsius and Fahrenheit readings equal?

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.

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?

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.

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?

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.

The number of bacteria in a culture doubles every 3 hours. Starting with 500 bacteria, how many will there be after 12 hours?

Answer: 8 000

After 12 hours, there are 12/3 = 4 doublings. Population = 500 × 2⁴ = 500 × 16 = 8 000 bacteria.

An investment of R10 000 earns compound interest at 8% per annum compounded quarterly. What is the value after 2 years?

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.