Question 344621: This is a calc. question; please help!
A telephone wire is to be laid from the telephone company to an island 7 mi off shore at a cost of $200,000 per mile along the shore line and $300,000 per mile under the sea. How should the wire be laid @ the least expensive cost if the distance along the shoreline in 12 miles. What is the cost?
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! A telephone wire is to be laid from the telephone company to an island 7 mi off shore at a cost of $200,000 per mile along the shore line and $300,000 per mile under the sea. How should the wire be laid @ the least expensive cost if the distance along the shoreline in 12 miles. What is the cost?
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This forms a right triangle with the shore side 12 miles long and the base = 7 miles to the island.
x = the distance along the shore from the perpendicular from the island.
The distance on land = 12 - x miles
The distance under water = sqrt(x^2 + 49) miles
The cost C = 200*(12-x) + 300*sqrt(x^2 + 49) (Cost in $1000)
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dC/dx = -200 + 300x*(x^2 + 49)^(-1/2) = 0
3x*(x^2 + 49)^(-1/2) = 2
3x = 2sqrt(x^2 + 49)
9x^2 = 4x^2 + 196
x^2 = 196/5
x = 2.8sqrt(5) miles for minimum cost
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Length of cable = 12 - 2.8sqrt(5) + sqrt(88.2)
L = 12 - 2.8sqrt(5) + 4.2sqrt(5)
L = 12 + 1.4sqrt(5) miles total
L =~ 15.13 miles
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Cost = 200*(12-x) + 300*sqrt(x^2 + 49) (Cost in $1000)
C = 2400 - 560sqrt(5) + 1260sqrt(5) = 2400 + 600sqrt(5)
Cost =~ $3,741,641
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