EXM.1.AHL.TZ0.21

pestleMathematicsAIHLPaper 1EXM· ahl-4-13-non-linear-regressionsource ↗

Saloni wants to find a model for the temperature of a bottle of water after she removes it from the fridge. She uses a temperature probe to record the temperature of the water, every 5 minutes.

After graphing the data, Saloni believes a suitable model will be

$T = 28 - a{b^t}$ , where $a{\text{,}}\,\,b \in {\mathbb{R}^ + }$ .

Explain why $28 - T$ can be modeled by an exponential function.

[1]

Find the equation of the least squares exponential regression curve for $28 - T$ .

[3]

Write down the coefficient of determination, ${R^2}$ .

[1]

c.i.

Interpret what the value of ${R^2}$ tells you about the model.

[1]

c.ii.

Hence predict the temperature of the water after 3 minutes.

[2]

Markscheme / solution

Rearranging the model gives $28 - T = a{b^t}$ A1

So $28 - T$ can be modeled by an exponential function. AG

[2 marks]

(A1)

$28 - T = 22.7{\left( {0.925} \right)^t}$ M1A1

[3 marks]

${R^2} = 0.974$ A1

[1 mark]

c.i.

Since the value of ${R^2}$ is close to +1, the model is a good fit for the data. R1

[1 mark]

c.ii.

$T = 28 - \left( {22.69 \ldots } \right){\left( {0.9250 \ldots } \right)^3} = 10.0$ minutes M1A1

[2 marks]

Examiners’ report

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c.i.

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c.ii.

[N/A]