14) A power line is to be installed across a river that is 1 mile wide to a town that is 5 miles...

Question

14) A power line is to be installed across a river that is 1 mile wide to a town that is 5 miles downstream from the power source on the other side of the river. (See diagram below.) It costs $7000 per mile to lay the cable underground and $9000 per mile to lay it underwater. Determine the distance in miles x the cable should run underground along the river from the power source and then be run underwater diagonally to the town if a total of $45,000 has been allocated and will be totally used for this project. (Show all work to justify your answer(s) for x and round your final answer(s) to two digits behind the decimal point and remember 0 5 x 5 5 miles. This is a lengthy problem so you can finish it on the next page.)

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David · Accepted Answer

Let the distance underground = x miles 
The total cost of wire underground = x*7000 
Thus total distance of wire under water = sqrt( (5-x)^2 + 1^2 )  
The total cost of wire underwater = sqrt( (5-x)^2 + 1^2 ) * 9000
Total cost = 45000 
45000 = sqrt( (5-x)^2 + 1^2 ) * 9000 + x*7000 
(45 – 7x)^2 = ((5-x)^2 + 1)*81 
2025 + 49x^2 – 630x = (25 + x^2 – 10x + 1)*81 
2025 + 49x^2 – 630x = (2025 +81x^2 – 810x + 81) 
32x^2 – 180x + 81  = 0 
Solving this quadratic equation we get: 
X = 5.1317 mile, 0.4933 mile 
Rejecting the value greater than 5 , we get x = 0.4933 mile

14) A power line is to be installed across a river that is 1 mile wide to a town that is 5 miles downstream from the power source on the other side of the river. (See diagram below.) It costs $7000...

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