To extend the model for the temperature M(t) over the full 0≤t≤25 minutes, we need to account for three distinct phases, as seen in the data and the graph:
Cooling to 0∘C (ice only): For 0≤t≤10, the original model C(t) accurately describes the temperature as the ice is being heated up to its melting point.
Melting phase (temperature held at 0∘C): For 10<t≤15, the temperature remains constant at 0∘C while the ice melts. Here, 15 is the time when all the ice has just finished melting.
Heating of water (after all ice has melted): For 15<t≤25, the temperature rises again as the water is heated. To model this, fit a new line through the point (15,0) and the measured temperature, approximately 5°C, at t=25. The equation of the line is:
where
and
This ensures the line passes through (15,0) and (25,5).
Graphically:
Plot C(t) from t=0 to t=10.
Draw a horizontal line at y=0 from t=10 to t=15.
From (15,0), draw the new heating curve (e.g., the straight line above) up to the point at t=25.
Combining these three segments gives a complete model M(t) for the temperature over the first 25 minutes.