Question
A small resort is situated on an island that lies exactly 6 miles from P, the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from P is the closest source of fresh water. If it costs 1.4 times as much money to lay pipe in the water as it does on land, how far down the shoreline from P should the pipe from the island reach land in order to minimize the total construction costs?
Answers
P=point on shore
R=resort
W=water source
X=point on shore where pipe enters water
Assume land pipe costs 1, water pipe costs 1.4
If X is at distance x from P, then
the cost of the pipeline
c = (10-x)+1.4sqrt(x^2+36)
dc/dx = -1 + 1.4x/sqrt(x^2+36)
dc/dx = 0 where
sqrt(x^2+36) = 1.4x
x^2+36 = 1.96x^2
.96x^2 = 36
x^2 = 37.5
x = 6.12
R=resort
W=water source
X=point on shore where pipe enters water
Assume land pipe costs 1, water pipe costs 1.4
If X is at distance x from P, then
the cost of the pipeline
c = (10-x)+1.4sqrt(x^2+36)
dc/dx = -1 + 1.4x/sqrt(x^2+36)
dc/dx = 0 where
sqrt(x^2+36) = 1.4x
x^2+36 = 1.96x^2
.96x^2 = 36
x^2 = 37.5
x = 6.12
Concur.
I tried this and it didn't work for my problem even though they were exactly the same except for the 1.4 in this problem being 2.9 in mine.
That would be because they missed the 2 in the denominator for their:
c = (10-x)+1.4sqrt(x^2+36)
dc/dx = -1 + 1.4x/sqrt(x^2+36)
it should actually be:
dc/dx = -1 + 1.4/2*sqrt(x^2+36)
c = (10-x)+1.4sqrt(x^2+36)
dc/dx = -1 + 1.4x/sqrt(x^2+36)
it should actually be:
dc/dx = -1 + 1.4/2*sqrt(x^2+36)