Posted: November 4th, 2015

Math Modeling

Math Modeling

1. (a) Find the curve with the property that light rays emanating from
x
y
<dx ,dy >
-a a
r?1 r?2
the point (-a, 0) are reflected to the point (a, 0).
Hint. This is similar to the parabola problem in class …
Use r?1 and r?2
(b) Show |r?1
| + |r?2
| is the same regardless of the path taken.
2. What is the solution of the brachistochrone problem if v =
q
-2g y 2
?
3. (a) Find ?(x ,t ) in the fan region for the initial condition
?(x , 0) =
?
?
?
?0
, x < 0
?0
(1 – x /L), 0 = x = L
0, x > L
using the quadratic velocity-density function u(?) = u0

1 – ?
2
/?2
0

, 0 = ? = ?0
.
(b) Put L = 0 in (a) and find the solution of the car equation dX /dt = u(?) in the fan.

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