written assignment 6 - JP Rodriguez Prof Muralee Calc 1 7...

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JP Rodriguez Prof Muralee Calc 1 7 December 2007 Written Assignment 6 A) A telephone wire is to be laid form location (a) on an island 7 miles off shore to a point (b) 12 miles down the coast. The cost of the wire is $2000/mile over land and $3000/mile across the ocean. What is the best way to plan the project so that cost is minimized? To complete this problem, we use optimization. First we set up an equation to find the cost of the wire over land, where x equals the distance over land, we get this: (2000)(12 - x ) Second, we find the cost of laying the wire across the sea, which is: (3000)(49 + x 2 ) 1/2 When we add these two formulas together, we can begin to find the cost of the project: C( x ) = 2000(12 – x ) + 3000(49 + x 2 ) 1/2 C( x ) = 2400 – 2000 x + 3000(49 + x 2 ) 1/2 Now we must find the derivative of the function, C’( x ): C’( x ) = -2000 + (1500/(49 + x 2 ) 1/2 )(2 x ) C’( x ) = (3000/(49 + x 2 ) 1/2 ) – 2000 We now take the derivative, set it equal to zero, and find the critical numbers: 2000(49 + x 2 ) 1/2 = 3000 x 2000 2 (49 + x 2 ) = 3000 2 x 2 196 x 10 6 + 4 x 10 6 x 2 = 9 x 10 6 x 2 196 x 10 6 = 5 x 10 6 x 2 x = 6.261 miles B)
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