Explanation Given: The outside temperature varies as s sine wave, with a minimum of 16 ° C at 2 : 00 A .M and maximum of 32 ° C at 2 : 00 P .M . Calculation: Consider the heat produced is at a constant rate H 0 and no heating and cooling that is, U ( t ) = 0 Consider the outside temperature M ( t ) varies as s sine wave over a 24 hrs period with its minimum at t = 0 and maximum at t = 12 therefore, M ( t ) = M 0 − B cos ω t Here, B is positive constant, and M 0 is the average outside temperature. And, ω = 2 π 24 = π 12 rad / hr Given that the outside temperature varies as s sine wave with a minimum of 16 ° C at 2 : 00 A .M and a maximum of 32 ° C at 2 : 00 P .M . Here, t = 0 represents 2 : 00 A .M , and t = 12 represents 2 : 00 P .M . Since, M 0 is the average outside temperature. M 0 = 32 + 16 2 = 24 Here, H 0 = 0 therefore, B 0 = 24 + 0 K = 24 Consider following the relationship. M ( t ) = M 0 − B cos ω t At t = 0 . M ( 0 ) = 24 − B cos 0 B = 24 − 16 = 8 Given that exponential term died off so, C = 0 The temperature function is given by, T ( t ) = M 0 − B 1 + ( ω K ) 2 sin ( ω t + tan − 1 ( K ω ) ) + ( 0 ) e − K t = M 0 − B 1 + ( ω K ) 2 sin ( ω t + tan − 1 ( K ω ) ) = 24 − 8 1 + ( ω K ) 2 sin ( ω t + tan − 1 ( K ω ) ) …… ( 1 ) The warehouse with no heating or cooling has a time constant of 1 hr , therefore, K = 1 As the maximum value of sine is 1

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Chapter 3.3, Problem 9E

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The highest and lowest temperature corresponding to the time constant of 1 hr and 5 hr.

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