anonymous
  • anonymous
An oil rig is to be built 30 miles down shore and 20 miles out to sea from a refinery. A pipe line needs to be built from the refinery to the oil rig. It costs $1000 per mile to build the pipe on land and $1500 per mile to build it at sea. The planners of this oil rig want to have the lowest cost possible for this pipe. How much would it cost to build the pipe line all the way down the shore until it is directly "below" the oil rig and then out to sea?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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amistre64
  • amistre64
are we assuming that the sea floor is flat?
anonymous
  • anonymous
yes
anonymous
  • anonymous
this starts out to be a minimization problem but then the question asks something different. confusing.

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amistre64
  • amistre64
10sqrt(13) * 1000 then
amistre64
  • amistre64
the minimal distance between 2 points on a plane is a stright line
amistre64
  • amistre64
its a pythag problem that has no real world application lol
amistre64
  • amistre64
it seems to be asking; how much to build a pipeline along the legs of the triangle tho doesnt it
anonymous
  • anonymous
not to me but my reading comprehension might be off.
amistre64
  • amistre64
1 if by land; 2 if by sea..... yeah, they want the minimal cost compared to yada yada yada
anonymous
  • anonymous
but it doesn't ask for minimum cost, even though it says "The planners of this oil rig want to have the lowest cost possible for this pipe."
anonymous
  • anonymous
is says "How much would it cost to build the pipe line all the way down the shore until it is directly "below" the oil rig and then out to sea? 8 minutes ago"
amistre64
  • amistre64
we have to determine at what cost the land pipe and the sea pipe are qual
amistre64
  • amistre64
yeah, below being a pictorial term
anonymous
  • anonymous
the shoreline is in a distance straight line from the refinery the oil rig is verticalto the shoreline and at angle between the shorline and refinery
anonymous
  • anonymous
it looks to me as if it is just asking how much it would cost to build the pipeline 30 miles down the shore and then 20 miles out to sea. that is what the last statement says unless i am reading it incorrectly. is this for a calc class?
amistre64
  • amistre64
25980.76 for a straight run to the oil rig
amistre64
  • amistre64
it is; but they want some variation of that;
anonymous
  • anonymous
30 miles down shore AND 20 miles out to sea from a refinery
anonymous
  • anonymous
i take that to mean east 30 and then 20 north, for example. but of course i could be wrong.
amistre64
  • amistre64
60000 if we build it at a right angle
amistre64
  • amistre64
52426.40 if we go with a 45 at 10 feet from the start
anonymous
  • anonymous
amistre64
  • amistre64
C(x) = s(1500) + l(1000)
amistre64
  • amistre64
how to relate s an l? by angle perhaps?
amistre64
  • amistre64
sin(t) = l cos(t) = 30 - cos(t) = s
amistre64
  • amistre64
if this was speed; then the fastest speed would be?
anonymous
  • anonymous
30mph
amistre64
  • amistre64
along which route :)
anonymous
  • anonymous
i dont understand your question?
amistre64
  • amistre64
straight shot; 54083.26 10 down and 45degrees gets us: 52426.40 11 down and a distance of sqrt(19^2 + 20^2): 52379.34 12(1000) + sqrt(18^2 + 20^2)(1500) = 52360.87
amistre64
  • amistre64
its a matter if finding the route by land and by sea that is the cheapest to build
amistre64
  • amistre64
30(1000) + 20(1500) = 60 000 ; so just building it up the 30 then 20 is bad
anonymous
  • anonymous
is sqrt supposed to mean log?
amistre64
  • amistre64
log means log sqrt means square root
anonymous
  • anonymous
what are you sqrting 11 down?
amistre64
  • amistre64
the pipeline goes along the beach; and each foot costs so much money ($1000) in that instance, we went down 11 feet and bent the pipe to go to the rig out at sea; the sea pipe costs 1500 per foot
amistre64
  • amistre64
ideally, when the land pipe cost matches the sea pipe cost we have the cheapest cost
amistre64
  • amistre64
when s(1500) = l(1000) we balance out and have the cheapest cost for total pipe
anonymous
  • anonymous
o i get it ......
anonymous
  • anonymous
from the refinery to the oil rig would be the hypotenous of the other sides? multiplied to get the cost?
amistre64
  • amistre64
yes
anonymous
  • anonymous
thank you soooooo very much for your help
amistre64
  • amistre64
can you get it from here?
anonymous
  • anonymous
yes i can thanks
anonymous
  • anonymous
i agree with amistre solution. however, since this is calc class you probably should (to make the teacher happy) have some sort of function to find the minimum of. so for the second part, and if you put "x" as suggested. then the distance from where the pipe enters the water to the oil rig is \[\sqrt{x^2+400}\] miles by pythagoras and the cost is $1500 per mile for a total of \[1500\sqrt{x^2+400}\] the distance from the refinery to where the pipe hits the water is 3\[30-x\] miles at a cost of $1000 per mile for a cost of \[1000(30-x)=30000-1000x\] and therefore your total cost will be \[C(x)=1500\sqrt{x^2+400}+30000-1000x\] this is the function you need to minimize
anonymous
  • anonymous
thanks
anonymous
  • anonymous
\[C'(x)=\frac{1500x}{\sqrt{x^2+400}}-1000\] set = 0 to find the critical point get \[x=8\sqrt{5}\]
anonymous
  • anonymous
and don't forget that the distance down the shore line is not x, but rather 30-x or about 12.11 rounded.
anonymous
  • anonymous
i got it lol
anonymous
  • anonymous
i did this on the fly so you should check my work. sure about the derivative though, and fairly positive about my \[x=8\sqrt{5}\]
anonymous
  • anonymous
ok
anonymous
  • anonymous
good luck.
anonymous
  • anonymous
thx
anonymous
  • anonymous
welcome
anonymous
  • anonymous
lol

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