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Unformatted text preview: Section 4.4
(b) The distance between the particles is the absolute value of f (t) sin t
3 163 42. The trapezoid has height (cos ) ft and the trapezoid bases measure 1 ft and (1 V( )
1 (cos )(1 2 sin t 3 2 cos t 1 sin t. Find 2 2 sin ) ft, so the volume is given by 1 2 sin )(20) the critical points in [0, 2 ]: f (t)
3 2 sin t 1 cos t 2 3 sin t 2 0
1 cos t 2 1 20(cos )(1 sin ).
2 Find the critical points for 0 V( ) 20(cos )(cos ) 20 cos2 20(1 sin2 ) 20(1 20 sin 20 sin : 0 0 0 0 0 1
6 sin )( sin ) 20 sin2 20 sin2 sin 1)(sin 1) 1) tan t 3 5 11 The solutions are t and t , so the critical 6 6 5 11 , 1 and , 1 , and the interval points are at 6 6 3 3 , and 2 , . The particles endpoints are at 0, 2 2 5 11 sec and at t sec, and are farthest apart at t 6 6 20(2 sin2 20(2 sin sin 1 or sin 2 the maximum distance between the particles is 1 m. The critical point is at (c) We need to maximize f (t), so we solve f (t) f (t)
3 2 0. 6 , 15 3 . Since V ( )
6 2 0 for 0 cos t 1 sin t 2 1 sin t 2 6 and V ( ) 0 for , the critical point 0
3 2 corresponds to the maximum possible trough volume. The cos t volume is maximized when 43. (a)
R D 8.5 y Q C S 6 . This is the same equation we solved in part (a), so the solutions are t
3 sec and t 4 sec. 3 For the function y , 1 and f (t), the critical points occur at
y L x x A P B 4 , 1 , and the interval endpoints are at 3 3 1 1 0, and 2 , . 2 2 4 Thus, f (t) is maximized at t and t . But 3 3 these are the instants when the particles pass each other, so the graph of y points and f (t) has corners at these Sketch segment RS as shown, and let y be the length of segment QR. Note that PB 8.5 x, and so QB x 2 (8.5 x)2 8.5(2x 8.5). Also note that triangles QRS and PQB are similar.
QR RS PQ QB x 8.5(2x 8.5) x2 8.5(2x 8.5) 8.5x 2 2x 8.5 d f (t) is undefined at these instants. We dt cannot say that the distance is changing the fastest at any particular instant, but we can say that near t t
3 4 the distance is changing faster than at any other 3 or y 8.5 y2 8.52 time in the interval. y2 L2 L2 L2 L2 x2 x2
x 2(2x y2
8.5x 2 2x 8.5 8.5) 8.5x 2 2x 8.5 2x 3 2x 8.5 ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN

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