SOLUTION: Power lines are going to be run from a power station on the shoreline to an island one mile off shore and 4 miles down the shore line as shown by the dotted line below. The cost t
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Question 1115085: Power lines are going to be run from a power station on the shoreline to an island one mile off shore and 4 miles down the shore line as shown by the dotted line below. The cost to run the lines over land is $2000 per mile and the cost to run the lines underwater is $6000 per mile. Let x represent the length of the part of the line that is run over land.
a) Express the length, L, of the part of the line that is run underwater as a function of x.
b) Express the cost, C, of the entire line as a function x.
c) Find a reasonable domain for the function found in part (b). Answer by greenestamps(13203) (Show Source):
The shoreline, the perpendicular distance of 1 mile from the shoreline to the island, and the path of the power line form a right triangle. The length of the side of that triangle along the shore is (4-x).
By the Pythagorean Theorem, the length of the part of the power line under water is
The total cost of the power line is $2000 for each mile along the shore and $6000 for each mile underwater:
It clearly would not make sense, if you were trying to minimize the total cost of the power line, to run the line more than 4 miles along the shoreline. So a reasonable domain would be x from 0 to 4.
