In the following diagram, S represents the position of a power relay station located on a straight coast, and E shows the location of a marine biology experimental station on an island. A cable is to be laid connecting the relay station with the experimental station. If the cost of running the cable on land is $1.50/running foot and the cost of running the cable under water is $2.70/running foot, locate the point P that will result in a minimum cost (solve for x). (Round your answer to the nearest whole number.)
Here is the figure - http://www.webassign.net/tanapmath5/10-5-025.gif
I'm really confused on the steps to do this!
Here is the figure - http://www.webassign.net/tanapmath5/10-5-025.gif
I'm really confused on the steps to do this!
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Distance from E to P = sqrt(3000^2 + x^2)
Cost = 2.7 * sqrt(3000^2 + x^2)
Distance from P to S = 10000 - x
Cost = 1.5 * (10000 - x)
Total cost = C = the sum of the two costs
C = 2.7 * sqrt(3000^2 + x^2) + 1.5 * (10000 - x)
C = 2.7 sqrt(x^2 + 3000^2) - 1.5 x + 15000
. . . C is minimized when C ' = 0
C ' = 2.7 x / (sqrt(x^2 + 3000^2)) - 1.5
2.7 x / (sqrt(x^2 + 3000^2)) - 1.5 = 0 <=== solve for x
x = (about) 2004.46 feet
Cost = 2.7 * sqrt(3000^2 + x^2)
Distance from P to S = 10000 - x
Cost = 1.5 * (10000 - x)
Total cost = C = the sum of the two costs
C = 2.7 * sqrt(3000^2 + x^2) + 1.5 * (10000 - x)
C = 2.7 sqrt(x^2 + 3000^2) - 1.5 x + 15000
. . . C is minimized when C ' = 0
C ' = 2.7 x / (sqrt(x^2 + 3000^2)) - 1.5
2.7 x / (sqrt(x^2 + 3000^2)) - 1.5 = 0 <=== solve for x
x = (about) 2004.46 feet