Modelling skills: fit, test, interpret, predict practice questions.

A linear model P = 3t + 45 gives a plant's height P (cm) after t weeks. Calculate the predicted height after 8 weeks.

• 53 cm
• 69 cm
• 24 cm
• 48 cm

Answer: 69 cm

Substitute t = 8: P = 3(8) + 45 = 24 + 45 = 69 cm.

A regression line for a data set has coefficient of determination R² = 0.81. State what proportion of the variation in the dependent variable is explained by the model.

• 9%
• 19%
• 81%
• 90%

Answer: 81%

R² = 0.81 means 81% of the variation in the response variable is explained by the linear model.

The exponential model N = 200(1.05)^t models a population after t years. Determine the initial population (t = 0).

• 210
• 200
• 205
• 1.05

Answer: 200

At t = 0, (1.05)^0 = 1, so N = 200(1) = 200.

A model cools a drink according to T = 20 + 60e^(−0.1t), where T is in °C and t in minutes. Determine the temperature the model predicts in the long term (as t → ∞).

• 80 °C
• 60 °C
• 20 °C
• 0 °C

Answer: 20 °C

As t → ∞, e^(−0.1t) → 0, so T → 20 + 60(0) = 20 °C, the horizontal asymptote (ambient temperature).

Two models are fitted to the same data. Model A has R² = 0.64 and Model B has R² = 0.92. Suggest which model gives the better fit.

• Model A, because R² is smaller
• Model B: R² is closer to 1
• Both fit equally well
• Cannot be judged from R²

Answer: Model B: R² is closer to 1

A higher R² (closer to 1) indicates the model explains more of the variation, so Model B fits better.

A quadratic model h = −5t² + 20t gives the height (m) of a ball after t seconds. Calculate the time when the ball returns to the ground (h = 0, t > 0).

• 2 s
• 4 s
• 5 s
• 20 s

Answer: 4 s

−5t² + 20t = 0 gives 5t(−t + 4) = 0, so t = 0 or t = 4. The positive non-zero solution is t = 4 s.