Modelling: linear, quadratic, exponential, direct/inverse variation, cubic, sinusoidal practice questions.

Solve the quadratic equation x² − 5x + 6 = 0.

• x = 1 or x = 6
• x = 2 or x = 3
• x = −2 or x = −3
• x = 1 or x = 5

Answer: x = 2 or x = 3

Factorising: (x − 2)(x − 3) = 0, so x = 2 or x = 3 (they sum to 5 and multiply to 6).

The height of a ball is modelled by h(t) = −5t² + 20t, with h in metres and t in seconds. Determine the maximum height reached.

• 20 m
• 15 m
• 25 m
• 40 m

Answer: 20 m

The maximum occurs at t = −b/(2a) = 20/10 = 2 s; h(2) = −5(4) + 40 = −20 + 40 = 20 m.

The value of a car is modelled by V(t) = 24000 × 0.85^t, with t in years. Calculate its value after 2 years, to the nearest dollar.

• $17 340
• $20 400
• $16 320
• $18 000

Answer: $17 340

V(2) = 24000 × 0.85² = 24000 × 0.7225 = 17 340.

The depth of water is modelled by d(t) = 6 + 3sin(30t°), with t in hours. State the minimum depth.

• 3
• 6
• 9
• 0

Answer: 3

sin ranges from −1 to 1, so the minimum of d is 6 + 3(−1) = 3.

A cubic model gives the volume of a box as V(x) = x(10 − 2x)(8 − 2x). Calculate V when x = 2.

• 48
• 24
• 96
• 12

Answer: 48

V(2) = 2 × (10 − 4) × (8 − 4) = 2 × 6 × 4 = 48.

The temperature of a cooling drink is modelled by T(t) = 20 + 60 × 0.5^t, with t in minutes. Determine the temperature it approaches after a long time.

• 20°C
• 80°C
• 0°C
• 60°C

Answer: 20°C

As t → ∞, 0.5^t → 0, so T → 20 + 0 = 20°C, the ambient temperature (horizontal asymptote).