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2. (a) A bowl of water has a temperature of 50 ◦ C. It is put into a refrigerator where the temper ature is 5 ◦ C. After 0.5 hours, the water is stirred and its temperature is measured to be 20 ◦ C. It is then left to cool further, take t in Newton’s law of cooling to be measured in minutes. Use Newton’s law of cooling to predict when the temperature will be 10 ◦ C. [15 marks] (b) Verify that the given family of functions solves the difffferential equation. (i) dy dt = (1 2t)y 2 , y = 1 C t + t2 . [10 marks] (ii) dy dt = y 2 sin t, y = 1 C + cost . [10 marks] (c) Evaluate the integral Z 1 0 x 2 (√4 x2 ) 3 dx [15 marks]
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