Pre-Calc Homework Solutions 199

Pre-Calc Homework Solutions 199 - Chapter 4 Review 52. 10 y...

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52. For 0 , x , } 5 3 p } , the area of the rectangle is given by A ( x ) 5 (2 x )(8 cos 0.3 x ) 5 16 x cos 0.3 x . Then A 9 ( x ) 5 16 x ( 2 0.3 sin 0.3 x ) 1 16(cos 0.3 x )(1) 5 16(cos 0.3 x 2 0.3 x sin 0.3 x ) Solving A 9 ( x ) 5 0 graphically, we find that the critical point occurs at x < 2.868 and the corresponding area is approximately 29.925 square units. 53. The cost (in thousands of dollars) is given by C ( x ) 5 40 x 1 30(20 2 y ) 5 40 x 1 600 2 30 ˇ x 2 w 2 w 1 w 4 w 4 w . Then C 9 ( x ) 5 40 2 } 2 ˇ x 2 w 30 2 w 1 w 4 w 4 w } (2 x ) 5 40 2 } ˇ x 2 w 3 2 w 0 x 1 w 4 w 4 w } . Solving C 9 ( x ) 5 0, we have: } ˇ x 2 w 3 2 w 0 x 1 w 4 w 4 w }5 40 3 x 5 4 ˇ x 2 w 2 w 1 w 4 w 4 w 9 x 2 5 16 x 2 2 2304 2304 5 7 x 2 Choose the positive solution: x 51} ˇ 48 7 w } < 18.142 mi y 5 ˇ x 2 w 2 w 1 w 2 w 2 w 5 } ˇ 36 7 w } < 13.607 mi 54. The length of the track is given by 2 x 1 2 p r , so we have 2 x 1 2 p r 5 400 and therefore x 5 200 2 p r . Then the area of the rectangle is A ( r ) 5 2 rx 5 2 r (200 2 p r ) 5 400 r 2 2 p r 2 ,for0 , r , } 2 p 00 } . Therefore, A 9 ( r ) 5 400 2 4 p r and A 0 ( r ) 52 4 p , so the critical point occurs at r 5 } 1 p 00 } m and this point corresponds to the maximum rectangle area because A 0 ( r ) , 0 for all r . The corresponding value of x is x 5 200 2 p 1 } 1 p 00 } 2 5 100 m. The rectangle will have the largest possible area when x 5 100 m and r 5 } 1 p 00 } m. 55. Assume the profit is k dollars per hundred grade B tires and 2 k dollars per hundred grade A tires.
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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.

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