Calculus problem, HELP please!?
In the following diagram, S represents the position of a power relay station located on a straight coast, and E shows the location of a marine biology experimental station on an island. A cable is to be laid connecting the relay station with the experimental station. If the cost of running the cable on land is $1.50/running foot and the cost of running the cable under water is $2.50/running foot, locate the point P that will result in a minimum cost (solve for x). (Round your answer to the nearest whole number.)
http://www.webassign.net/tanapmath5/10-5-025.gif <--------- DIAGRAM
Comments
you can muck around with trig for these, but that is not fun.
Best to do this with geometry
the length of the diagonal line running over to p can be found using Pythagoras
p^2 = x^2 + 3000^2
p = rt(x^2 + 3000^2)
p costs 2.5 a foot.
the cost then is 2.5p
the cost for running along the shore is 1.5(10000-x)
add them together
C = 2.5p + 1.5(10000-x)
C = 2.5rt(x^2 + 3000^2) + 1.5(10000-x)
C = 2.5(x^2 + 3000^2)^(1/2) + 1.5*10000 - 1.5x
take the derivative and set equal to zero
C' = 2.5(1/2)(x^2+3000^2)^(-1/2)(2x) - 1.5 = 0
2.5x/rt(x^2 + 3000^2) = 1.5
(5/2)x = (3/2)rt(x^2 + 3000^2), square both sides
25/4x^2 = (9/4)(x^2 + 3000^2)
25x^2 = 9x^2 + 9*3000^2
16x^2 = 9*3000^2
x^2 = 9*3000^2/16
x = rt(9*3000^2/16)
x = 9000/4 = 2250 feet