(a) Interpretation of 25 in the context

The value of 25 represents the room temperature, which is the lowest temperature that the hot water will cool down to. It acts as the limiting temperature of the system.

(b) Finding the values of \( a \) and \( b \)

Using a graphic display calculator (GDC) to fit the power model:

• First, subtract 25 from all temperature values in the data.
• Perform a regression analysis to fit \( T(t) = at^b + 25 \).
• Obtained values: \( a = 244 \) (243.920…) and \( b = -1.03 \) (-1.02965…).

(c) Finding the values of \( k \) and \( c \)

For Soo Min’s exponential model \( T(t) = kc^t + 25 \), we perform exponential regression:

• Subtract 25 from the temperature values.
• Fit an exponential regression model to the data.
• Obtained values: \( k = 61.1 \) (61.0848…) and \( c = 0.923 \) (0.923029…).

(d) Justification for Soo Min’s model being a better fit

Soo Min’s model is a better fit because:

• Comparing \( R^2 \) values: The coefficient of determination \( R^2 \) for Akira’s model is approximately 0.7657, whereas for Soo Min’s model, it is approximately 0.9233. A higher \( R^2 \) value indicates a better fit.
• Physical interpretation: The exponential model better represents the cooling process because water temperature decreases exponentially rather than following a power-law function.
• Initial condition at \( t = 0 \): Soo Min’s model does not predict a temperature exceeding 100°C at \( t = 0 \), which is more physically realistic.

…………………………..Markscheme…………………………..

(a)

• Correctly identifying 25 as the room temperature or the minimum cooling temperature.

(b)

• Evidence of subtracting 25 from the temperature values.
• Correct regression values: \( a = 244 \), \( b = -1.03 \).

(c)

• Correct regression values: \( k = 61.1 \), \( c = 0.923 \).

(d)

• Valid comparison of \( R^2 \) values, showing Soo Min’s model is more accurate.
• Explanation based on the physical cooling process and better model representation.