A lighthouse, L, is located on a small island 4 km west of point A on a straight north-south coastlne. A power cable is to be laid from L to the nearest source of power at point B on the shoreline, 12 km north of point A. The cost of laying cable under water is $6000/km and the cost of laying cable along the shoreline is $2000/km. To minimize the cost, the power line will be built from L underwater to a point C on the shoreline and then alone the shoreline from C to B. Find the location of point C (to the nearest metre) on the shoreline where the power cable should enter the water.
so far, I am at the part to find the domain of x.
At this point, I believe that the domain of x is...
However, I am not 100% sure on this..
When I did proceed with this domain I found that the min cost was 48,000 dollars per km when the cable is 12 km on the shoreline and 4 km underwater
I dont know if this is correct.. can someone verify with me?
Yes on this account.At this point, I believe that the domain of x is
Drawing a diagram IS very helpful, especially when they tell you how they want the diagram to be drawn. If we have the LH to the west of A, and a point C between A and B (they tell us the best way to minimize our costs are if we run a cable underwater to the shore between A and B), we have a triangle:
We can call the hypotenuse: |LC| which is the distance from the lighthouse to the point C between A and B
We can call the base: |LA| defined as the shortest from the lighthouse to the shore, which is 4KM
We can call the side opposite the Lighthouse: |AC|, which we can label X, which is the distance from A to C, and is the POINT on the shore where our underwater cable ends
Above the Triangle LAC we have a single north-south line:
We can call the distance from B to the point C: |BC| which is the point B to C, which is the cabling for the ground - we know the maximum this length can be is 12 (not really since we are told to get a point between A and B), and that any other point would be 12-X
The next step would be to determine just what the relationships are between these sides/lines. From there, we can then form a function that gives us the cost of the underwater cable (the hypotenuse of our triangle, |LC|), and the shoreline-cable (|BC|).
No. Using your measurements, that would mean they lay the cable underwater in a straight line from the Lighthouse to the point A, and then lay shoreline cable from A to B. Draw that on a piece of paper; you will see you get a right triangle. What is smaller the Hypotenuse, or the 12KM+4KM?
$48,000 is going to be your maximum cost, but we want the OPTIMIZATION of this situation; in that we want the MINIMUM cost, and the corresponding distance of C from A that relates to that.
The distance I got from A to C was . Hopefully I did that right, and perhaps someone can hop in and check.