In a chemical reaction, a chemical mixture is left to cool down to room temperature. The temperature

Question

In a chemical reaction, a chemical mixture is left to cool down to room temperature. The temperature of the chemical mixture, $\theta {{\ }^{{}^\circ }}\mathrm{C}$ at time $t$ minutes is given in this table.

Time ($t\text{ minutes}$)	$1$	$6$	$12$	$18$	$24$	$30$
Temperature ($\theta\,^\circ\mathrm{C}$)	$97$	$62$	$45$	$37$	$31$	$27$

Sketch a scatter diagram for the data given in the table.[2]

A student attempts to model the relationship between $\theta $ and $t$ using two models:

(A) $\theta =a+bt$ or (B) $\theta =c+\frac{\mathrm{d}}{t}$

By calculating the product moment correlation coefficients and referring to the scatter diagram, explain clearly which of the two models is a more appropriate model for this set of data.[2]
Use the model you identified in (b) to find the equation of a suitable regression line and use your equation to estimate to predict the temperature when the time is $45$ minutes. Comment on the reliability of the estimate.[3]

A temperature of $T$ Kelvins is equivalent to a temperature of $\theta $ degrees Celsius, where $T=\theta +273.15$.

Re-write your equation in (c) so that it can be used when the temperature of the chemical, $\theta $, is given in Kelvins.[1]

Accepted Answer

(b) For model A, ${r = -0.909}$. For model B, ${r = 0.933}$ Model B is more appropriate since its ${\left\vert{}right\vert{}}$ is closer to ${1}$ and the scatter diagram shows that a curve fits the data points better than a straight line. (c) ${	heta = 36.4^\circ	ext{C}}$ (${3}$ s.f.) (d) ${T = 308 + \left(\frac{64.4}{t}ight)}$

2023 ASRJC P2 Q6

2023 ASRJC P2 Q6

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