1205finalF2006Key - Name ANSWER KEY ID MATH 1205 CRN FINAL...

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Name: ANSWER KEY MATH 1205 FINAL Fall, 2006 ID #: CRN: (50 pts total) Free Response Questions Read the Directions. CALCULATORS MAY BE USED You must SHOW ALL WORK and use methods learned in class to receive full credit. 1. (8 pts) A manufacturer wishes to minimize the cost of materials to produce 1 liter (=1000 cm^3) cans (cylindrical). Assuming the thicknesses of the ends and the side are the same, what is the ratio of the height to the diameter of the can that minimizes cost? V = 1000 cc = ! r 2 h h = 1000 ! r 2 minimize Surface Area ratio of ht to diameter SA = 2 ! r 2 + 2 ! rh = ! r 2 + 2 ! r 1000 ! r 2 " # $ % & SA ( r ) = 2 ! r 2 + 2000 r ( 1 S ) A = 4 ! r ( 2000 r ( 2 = 4 ! r 3 ( 2000 r 2 = 0 / undefined 4 ! r 3 ( 2000 = 0 * CN : r = 500 ! 3 TEST : S !! A = 4 " + 4000 r # 3 S !! A r = 500 " 3 $ % & ( ) > 0 *+ minumum h = 1000 " 500 " 3 $ % & ( ) 2 = 1000 " 500 2 3 " 2 3 $ % & ( ) = 1000 500 2 3 " 1 3 Ratio of height to diameter: 1000 500 2 3 " 1 3 2(500 1 3 ) " 1 3 = 1000 500 2 3 " 1 3 $ % & ( ) " 1 3 2(500 1 3 ) $ % & ( ) = 500 500 = 1 1 Ratio of height to diameter: 1 to 1 2. (7 pts) Given f ( x ) = ! 4( x 2 ! x ! 2) x 2 ! 5 x + 6 . Evaluate the following limits. State + ! or " ! where appropriate.
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