NBT Mathematical Modelling 2

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?

Accepted Answer

30 m × 60 m

Answer

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?

Accepted Answer

-40°

Answer

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?

Accepted Answer

2 000 or 10 000 units

Answer

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?

Accepted Answer

100 minutes

Answer

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?

Accepted Answer

8 000

Answer

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

(NBT) National Benchmark Test Practice Test

(NBT) National Benchmark Test Practice Test

NBT Mathematical Modelling 2