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:

1. 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.

   ​

   M(t)=C(t)

   ​

2. 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.

   ​

   M(t)=0

   ​

3. 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:

   ​

   M(t)=mt+b

   ​

   where

   ​

   m=25−155−0​=21​

   ​

   and

   ​

   b+15m=0⟹b=−15m=−215​

   ​

   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.

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:

1. 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.

   ​

   M(t)=C(t)

   ​

2. 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.

   ​

   M(t)=0

   ​

3. 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:

   ​

   M(t)=mt+b

   ​

   where

   ​

   m=25−155−0​=21​

   ​

   and

   ​

   b+15m=0⟹b=−15m=−215​

   ​

   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.