Pressure–Volume Power Model

00:00 7 marks

The relationship between the pressure (P kPa) and volume (V m³) of a gas at constant temperature is investigated. The following data is collected:

Volume (V m³)	Pressure (P kPa)
0.5	202
1.0	101
1.5	68
2.0	50
2.5	41
3.0	34

It is believed this relationship can be modelled by an equation of the form \(P = aV^b\).

(a) By using logarithms to linearize the data, find the value of b. [4]
(b) Find the value of a. [1]
(c) Find the value of the coefficient of determination \(R^2\) for the non-linear model \(P = aV^b\). [2]

Part a

4 minutes 4 marks

Question

By using logarithms to linearize the data, find the value of b. [4]

Hint Hint

Hint

Form \(\ln P=\ln a + b\ln V\) and regress ln P on ln V.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation

Answer Answer

Complete Answer

b \approx -0.995

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Part b

1 minutes 1 marks

Question

Find the value of a. [1]

Hint Hint

Hint

Use the intercept from the regression: a = e^(intercept).

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation

Answer Answer

Complete Answer

a \approx 101

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Part c

2 minutes 2 marks

Question

Find the value of the coefficient of determination R² for the non-linear model P = aV^b. [2]

Hint Hint

Hint

Read R² directly from the regression output.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation

Answer Answer

Complete Answer

R² ≈ 1.00

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation