Differential equations.. Need help with parts (c) & (d) please! 7. (IO) Suppose

Question

Differential equations.. Need help with parts (c) & (d) please!

7. (IO) Suppose the outside ternperature M(t) varies as a sine wave over a 24-hr period, with its minimum at midnight (t 0), awd its max- imum at noon (t 12). Then M(t) Mo A. wt m/2) where Mo is the average outside temperature, A is a positive constant and K/12. (a) Use a trigonometric identity to write M(t) in a simpler form (b) What do Mo A and Mo A represent? (c) A building with no heating or cooling has 10 a time constant of 2 hr. Find the temper- ature of the building T (t) if the outside temperature varies as a sine wave with a minimum of 50 F at midnight, and a maximum of 800 F at noon. (d) Determine the times at which the build- ing reaches its lowest temperature and its highest temperature, assuming the expo- nential term has died off. You must jus- tify your answer analytically.

Explanation / Answer

M(t) = M0+A sin (pit/12 - pi/2)

= M0-A cos pit /12

M0+A = Maximum temperature when t = 18

M0-A = Minimum temp when t=0

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c) Min = 50 and max = 80

Average = 65

M0 = 65

A = 80-65 = 15

Here 2 hours instead of 24 hours.

M(t) = M0+A sin (pit/12 - pi/2)

= M0-A cos pit /12

= 65-15 cos pit/12

Lowest is when t = 0 and minimum = 50

Highest when t = 18

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