$\"\"$

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(a)

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The initial temperature of the coffee is $\"\"$ .

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Room temperature is $\"\"$ .

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After $\"\"$ minutes, temperature is $\"\"$ then $\"\"$ .

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Newton $\"\"$ s Law of Cooling is $\"\"$ .

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The change of temperature : $\"\"$ .

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Here $\"\"$ is the temperature of the object and $\"\"$ is the surrondings temperatutre.

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At intital time : $\"\"$ .

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$\"\"$ .

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The general equation is $\"\"$ .

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ in the equation $\"\"$ .

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$\"\"$ .

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Temperature after $\"\"$ minutes is $\"\"$ .

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Substitute $\"\"$ in the above equation.

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in the equation $\"\"$ .

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$\"\"$

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From the equation $\"\"$ .

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$\"\"$

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$\"\"$ .

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Temperature of the coffee at time $\"\"$ minutes is $\"\"$ .

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$\"\"$

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(b).

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The average value of temperature during the half an hour.

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The integral limits are $\"\"$ to $\"\"$ .

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The average function is $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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The average temperature in first half an hour is $\"\"$ .

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$\"\"$

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(a) Temperature of the coffee at time $\"\"$ minutes is $\"\"$ .

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(b) The average temperature in first half an hour is $\"\"$ .