Calculus Question

Question

A small resort is situated on an island that lies exactly 5 miles from 𝑃P, the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from 𝑃P is the closest source of fresh water. If it costs 2.4 times as much money to lay pipe in the water as it does on land, how far down the shoreline from 𝑃P should the pipe from the island reach land in order to minimize the total construction costs? 

Answer 1

we seek to minimize 2.4 sqrt(5^2+x^2) + abs(10-x)

The derivative is 2.4x/sqrt(25+x^2) - 1 for 0 <= x <= 10

Then, 2.4x/sqrt(25+x^2) = 1, or

sqrt(25 + x^2) = 2.4x, or

25 + x^2 =5.76x^2

25 = 4.76x^2 = 119/25 x^2

x = 25 sqrt(119)/119 or, rounded to 6 significant digits, 2.291746