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Unformatted text preview: 166 Section 4.4
0. 0.
a 3 6 a 3 , the relationship is 3 53. (a) According to the graph, y (0) (b) According to the graph, y ( L) (c) y(0) 0, so d 0. 2bx (c) Since r
r h and h 2. Now y (x) c 3ax 2 c, so y (0) implies that ax 3 bx 2 and aL 3 bL 2 H 55. (a) Let x0 represent the fixed value of x at point P, so that P has coordinates (x0, a), and let m f (x0) be the slope of line RT. Then the 0. Therefore, y(x) 3ax 2 y (x) 2bx. Then y( L) 3aL 2 2bL and y ( L) 0, so we have two linear equation of line RT is y m(0 m(x x0) x0) a a. The yintercept of this line is a mx0,
mx0 m a equations in the two unknowns a and b. The second
3aL . Substituting into the first 2 3aL 3 aL 3 equation, we have aL 3 H, or H, so 2 2 H H a 2 3 . Therefore, b 3 2 and the equation for y is L L H H x 3 x 2 y(x) 2 3 x 3 3 2 x 2, or y(x) H 2 3 . L L L L equation gives b and the xintercept is the solution of m(x x0) a 0, or x . Let O designate the origin. Then (Area of triangle RST) 2(Area of triangle ORT) 2 2 m m
1 (xintercept of line RT)(yintercept of line RT) 2 1 mx0 a 2 mx0 m mx0 m a 2 m a 2 m a 54. (a) The base radius of the cone is r height is h Therefore, V(x)
3 a2 r2
2 a 2 x 2 a2 a2 2 a x and so the 2 2 a x 2 . 2 2 a 2 x 2 r 2h 3 . (a mx0)
a (b) To simplify the calculations, we shall consider the volume as a function of r: volume f (r) = f(r)
d 2 (r 3 dr 3 3 2a 2r 3 r(2a a2
2 2 mx0 m 3 r2 a2 r 2) r 2, where 0 r a. m x0 a2
1 Substituting x for x0, f (x) for m, and f (x) for a, we ( 2r) ( a2 r 2)(2r) have A(x) f (x) x
f(x) 2 . f (x) r2
2 r
3 a2 2r(a a
2 2 r2 r
2 r 2) (b) The domain is the open interval (0, 10). To graph, let y1 y2
2 3r 3 r2 3r 2) r
2 f (x) 5 5 1 x2 , 100 f (x) A(x) NDER(y1), and y2 x
y1 2 y2 3 a The critical point occurs when r r h r
a 3 2 a 3 a 6 . Then 3 2a 2 , which gives 3 y3 . A(x) is shown The graph of the area function y3 below. a2
6 r2 and h a2
a 3 , 3 2a 3 2 a 3 2 a 3 3 . Using we may now find the values of r and h for the given values of a.
[0, 10] by [ 100, 1000] When a when a when a when a 4: r 5: r 6: r 8: r 4 6 ,h 3 5 6 ,h 3 4 3 5 3 3 3 ; ; 2
8 3 6, h
6 2 3;
8 3 3 The vertical asymptotes at x 0 and x 10 correspond to horizontal or vertical tangent lines, which do not form triangles. ,h ...
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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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