ps02sol

# ps02sol - 18.03 Problem Set 2 Fall 2009 Solutions 1(15...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 18.03 Problem Set 2 Fall 2009 Solutions 1. (15 points) This problem is based on the example on pages 57–58 of EP. Suppose that the outdoor temperature in Cambridge during September is given by A ( t ) = 70- a cos( πt/ 12) . ( OUTSIDE ) You can think of A as standing for “ambient.” Here a is a positive constant, and t is the time in hours after midnight (on September 1, say). This means that the temperature oscillates daily with a minimum value of 70- a at midnight and a maximum value of 70 + a at noon. Your dormitory room has no source of heat or cooling, so according to Newton’s law of cooling, the temperature in your room satisfies the differential equation dT ( t ) dt =- k ( T ( t )- A ( t )) . ( ODE ) Here k is a positive constant that depends on how well-insulated your dorm room is. a) If you insulate your room better, will k increase or decrease? Explain why. If you insulate your room better, k decreases. When k is a small positive constant, the indoor temperature changes at a rate which is only a small amount times the difference in indoor/outdoor temperatures; this corresponds to good insulation. (In the limiting case of k = 0, we have dT/dt = 0, the case of perfect insulation, where the indoor temperature doesn’t depend on the outside temperature.) When k is large, the indoor temperature changes quickly in response to the temperature difference; this corresponds to poor insulation. (In the limiting case of infinite positive k , the indoor temperature would instantaneously equalize with the outdoors.) b) The text says that “typical” values of k might range from . 2 to . 5 . How would these values change if we used Celsius instead of Fahrenheit? What about Kelvin? What if we measured t in days instead of hours? Celsius is Fahrenheit translated and rescaled. Thus T ( t )- A ( t ) is rescaled. However, dT ( t ) /dt is also rescaled (and the derivative ignores the translation), and so k is unaffected....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

ps02sol - 18.03 Problem Set 2 Fall 2009 Solutions 1(15...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online