2011_Solution_7 - MA 1505 Mathematics I Tutorial 7...

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MA 1505 Mathematics I Tutorial 7 Solutions 1. (i) As the temperature function is only valid within the hotel room, its domain is { ( x, y ) : 0 x 10 , 0 y 10 } . (ii) The heater is at the location where the temperature is highest. It is clear that the largest value of T ( x, y ) is 36 at (0 , 5). (iii) The level curve of 20 is x 2 + ( y - 5) 2 = 80 which is an arc of the circle centered at (0 , 5) with radius 80 8 . 9. The level curve of 25 is x 2 + ( y - 5) 2 = 55 which is an arc of the circle centered at (0 , 5) with radius 55 7 . 4. So the bed should be placed somewhere between the two arcs. (iv) The level curves of c is x 2 + ( y - 5) 2 = 5(36 - c ). These are circles centered at (0 , 5), the values of c decreasing as the radius increases. The largest circle intersecting the domain intersects the domain at (10 , 0) and (10 , 10), so these points have the lowest temperature which is 11. 2. V = I × R = I = V R . (i) ∂I ∂V = 1 R . If R = 15, then ∂I ∂V = 1 15 0 . 0667 A/V. (ii) ∂I ∂R = - V R 2 . If V = 120 and R = 20, then ∂I ∂R = - 120 20 2 = - 0 . 3 A/Ω.
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